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THE  SUMARIO  COMPENDIOSO 


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THE  SUMARIO  COMPENDIOSO 
OF  BROTHER  JUAN  DIEZ 


THE  EARLIEST  MATHEMATICAL  WORK 
OF  THE  NEW  WORLD 


BY 


DAVID  EUGENE  SMITH 


BOSTON  AND   LONDON 
GINN  AND  COMPANY,  PUBLISHERS 

MDCCCCXXI 


COPYRIGHT,  1921,   BY  DAVID  EUGENE  SMITH 

ALL    RIGHTS    RESERVED 


THE  ATHF.N;€:UM   PRESS  •  CAMBRIDGE  •  MASSACHUSETTS  •  U.S.A. 


PREFACE 

If  the  student  of  the  history  of  education  were  asked  to  name  the  earUest  work 
on  mathematics  published  by  an  American  press,  he  might,  after  a  little  investiga- 
tion, mention  the  anonymous  arithmetic  that  was  printed  in  Boston  in  the  year  1729. 
It  is  now  known  that  this  was  the  work  of  that  Isaac  Greenwood  who  held  for  some 
years  the  chair  of  mathematics  in  what  was  then  Harvard  College.  If  he  should 
search  the  records  still  farther  back,  he  might  come  upon  the  American  reprint  of 
Hodder's  well-known  English  arithmetic,  the  first  textbook  on  the  subject,  so  far 
as  known,  to  appear  in  our  language  on  this  side  the  Atlantic.  If  he  should  look 
to  the  early  Puritans  in  New  England  for  books  of  a  mathematical  nature,  or  to 
the  Dutch  settlers  in  New  Amsterdam,  he  would  look  in  vain ;  for,  so  far  as 
known,  all  the  colonists  in  what  is  now  the  United  States  were  content  to  depend 
upon  European  textbooks  to  supply  the  needs  of  the  relatively  few  schools  that 
they  maintained  in  the  seventeenth  century. 

The  earliest  mathematical  work  to  appear  in  the  New  World,  however,  ante- 
dated Hodder  and  Greenwood  by  more  than  a  century  and  a  half.  It  was  published 
long  before  the  Puritans  had  any  idea  of  migrating  to  another  continent,  and  fifty 
years  before  Henry  Hudson  discovered  the  river  that  bears  his  name.  Of  this 
work  there  remain  perhaps  only  four  copies,  and  it  is  desirable,  not  alone  because 
of  its  rarity  but  because  of  its  importance  in  the  history  of  education  on  our  con- 
tinent, that  a  record  of  the  text  should  be  made  generally  accessible. 

In  making  the  translation  the  original  methods  of  expression  have  been  fol- 
lowed in  many  cases  in  which  smoother  diction  would  have  suggested  greater  free- 
dom of  rendition.  The  reason  has  been,  in  general,  the  desire  to  bring  such 
expressions  as  "  8  per  cent,"  '"  8  per  100,"  "  one  cosa  and  |  di^cosa,"  and  "four 
and  i"  into  sharp  contrast  with  the  corresponding  ones  of  the  present  time,  and 
to  show  the  great  advance  in  symbolism  and  in  methods  of  solution  and  proof. 
Such  minor  errors  as  were  common  in  a  century  of  careless  proofreading  have 
usually  been  corrected  without  comment.  Aside  from  this,  a  certain  freedom  of 
translation  has  been  assumed  for  the  evident  purpose  of  aiding  the  reader  to 
follow  the  spirit  of  the  text. 

The  editor  wishes  to  express  his  indebtedness  to  Sefiorita  Carolina  Marcial 
Dorado  for  her  scholarly  assistance  in  the  translation  of  the  Spanish  text. 

DAVID  EUGENE  SMITH 


CONTENTS 

PAGE 

INTRODUCTION i 

THE  MEXICO  OF  THE  PERIOD 3 

PRINTING  ESTABLISHED  IN  MEXICO 5 

GENERAL  DESCRIPTION  OF  THE  BOOK 7 

NATURE  OF  THE  TABLES 9 

THE  TEXT 13 

INDEX 65 


INTRODUCTION 


THE  MEXICO  OF  THE  PERIOD 

In  order  to  understand  the  Sumario  Compendioso  it  is  necessary  to  consider 
briefly  the  poHtical  and  social  situation  in  Mexico  in  the  middle  of  the  sixteenth 
century.  Cortes  entered  the  ancient  city  of  Tenochtitlan,  later  known  as  Mexico, 
in  the  year  15 19,  but  its  capture  and  destruction  occurred  two  years  later,  in  1521. 
Thus,  in  the  very  year  that  Luther  was  attacking  certain  ancient  privileges  in  the 
Old  World,  the  representatives  of  other  ancient  privileges  were  attacking  and 
destroying  a  worthy  civilization  in  the  newly  discovered  continent. 

The  rebuilding  of  the  city  began  at  once,  and  the  new  capital  soon  entered 
upon  an  era  of  great  prosperity,  disturbed,  however,  by  the  failure  of  Cortes  to 
show  the  power  as  a  civil  leader  that  he  had  shown  in  his  military  capacity.  The 
first  viceroy  of  New  Spain,  which  included  the  present  Mexico,  was  a  man  of 
remarkable  genius  and  of  prophetic  vision,  —  Don  Antonio  de  Mendoza.  He 
assumed  his  office  in  1535,  and  for  fifteen  years  administered  the  affairs  of  the 
colony  with  such  success  as  to  win  for  himself  the  name  of  "  the  good  viceroy." 
He  founded  schools,  established  a  mint,  ameliorated  the  condition  of  the  natives, 
and  encouraged  the  development  of  the  arts.  In  his  efforts  at  improving  the  con- 
dition of  the  people  he  was  ably  assisted  by  Juan  de  Zumarraga,  the  first  Bishop 
of  Mexico.  Among  the  various  activities  of  these  leaders  was  the  arrangement 
made  with  the  printing  establishment  of  Juan  Cromberger  of  Seville  whereby  a 
branch  should  be  set  up  in  the  capital  of  New  Spain. 

Mendoza  became  viceroy  of  Peru  in  1550  and  died  in  Lima  in  1552.  Upon 
leaving  Mexico  he  was  succeeded  by  Don  Luis  de  Velasco,  a  member  of  an  illus- 
trious Castilian  family  and  one  who  labored  faithfully  for  the  betterment  of  the 
people  intrusted  to  his  charge.  One  of  the  first  steps  taken  by  him  was  to  found, 
in  I55i>  the  Real y  Pontificia  Universidad  de  la  Ciudad  de  Mexico,  and  he  was 
at  all  times  interested  in  the  success  of  the  press  and  in  the  work  of  its  manager, 
Juan  Pablos,  as  the  name  appears  in  the  books  of  the  period.  Don  Luis  died  in 
1 564,  sincerely  mourned  by  the  people,  and  was  laid  to  rest  in  the  Monastery  of 
Santo  Domingo,  in  the  city  in  which  he  had  exercised  his  benevolent  authority. 

In  the  same  year  that  Mendoza  left  Mexico  for  Peru,  Zumarraga  passed  away, 
his  death  being  a  genuine  loss  to  the  State  as  well  as  the  Church.  In  the 
following  year  Alonso  de  Montufar*  was  nominated  as  his  successor  and  was 

*  The  spelling  is  substantially  as  given  in  the  facsimile  on  page  62.  The  first  name  usually 
appears  as  Alphonso  or  Alfonso. 


consecrated  in  1553.  For  sixteen  years  he  presided  with  great  success  over  the 
Church  in  New  Spain,  and  five  years  after  the  death  of  Don  Luis  de  Velasco  he 
too  was  buried  within  the  precincts  consecrated  to  the  memory  of  Santo  Domingo. 
It  was  in  his  time  and  with  his  sanction  that  the  Sumario  Compendioso  was  issued 
from  the  press,  and  well  did  he  deserve  the  praise  accorded  to  him  by  a  con- 
temporary writer  in  the  words 

"Clarissimo,  et  omnibus  animi  bonis  omatissimo  Sacr.  Theolog.  Mag.  Fr. 
Alphonso  d  Montufare,  Archipraesuli  Mexicano." 

Tempora  mutantur,  nos  et  mutamur  in  illis.  Such  words  of  praise,  expressed 
in  the  most  sonorous  of  tongues  and  in  words  that  seem  exaggerated  to  our  ears, 
belong  to  the  past.  In  our  rapid,  unsettled,  materialistic  life  we  seem  to  take 
pride  in  our  neglect  of  dignity  of  eulogy,  excusing  ourselves  by  condemning  the 
past  as  insincere  in  its  praise.  Who  that  reads  the  story  of  this  early  period, 
however,  can  say  that  such  descriptions  of  the  characters  and  accomplishments 
of  those  who  carried  the  Cross  to  the  New  World  were  exaggerated,  or  that  they 
failed  to  express  the  genuine  sentiments  of  the  people  to  whose  spiritual  needs 
these  brothers  of  the  holy  orders  so  conscientiously  ministered  ? 


II 

PRINTING  ESTABLISHED  IN  MEXICO 

The  idea  of  setting  up  a  press  in  Mexico  seems  to  have  been  considered  as 
early  as  1534,  even  before  Mendoza  became  viceroy,  doubtless  at  the  suggestion 
of  Juan  de  Zumarraga;  but  it  was  not  until  1536  that  the  plan  was  carried  out, 
Juan  Cromberger  then  sent  over  as  his  representative  Juan  Pablos,  a  Lombard 
printer,  and  so  the  "casa  de  Juan  Cromberger"  was  established,  prepared  to  spread 
the  doctrines  of  the  Church  to  the  salvation  of  the  souls  of  the  unbelievers. 
Cromberger  himself  never  went  to  Mexico,  but  his  name  appears  either  on  the 
portadas  or  in  the  colophons  of  all  the  early  books.  From  and  after  1545,  however, 
the  name  is  no  longer  seen,  Cromberger  having  died  in  1540. 

It  was  this  John  Paul  who  printed  the  Sumario  Compendioso,  in  1556,  and  in 
order  that  the  significance  of  the  work  may  be  the  better  appreciated  it  is  appropriate 
to  mention  the  following  books,  known  to  have  been  printed  by  him  before  that  year  : 

c.  1537.    Escala  Espiritual para  llegar  al  ctelo,  possibly  in  1537,  but  there  is  no  copy  extant. 

1 539.  Breve  y  mds  compendiosa  doctrina  Christiana  en  lengua  Mexicana  y  Castellana. 

1540.  Manual  de  Adultos. 

1 54 1 .  Relacion  d'l  espdtable  terremoto. 

1543.  Doctrina  breue. 

1 544.  Tripartita  del  Christianissimo  y  consolatorio  doctor  Juan  Gerson.  This  contains  the 
earliest  woodcut  printed  in  Mexico. 

1 544.    Copedio  breue  .  .  .     j>cessiones  .  .  .  por  Dionisio  Richel. 
c.  1 544.  A  second  edition  of  the  preceding  work. 
1 544.    Dotrina  xpiana. 
1 545- 1 546.    Doctrina  cristiana. 
1546.    Doctrina  xpiana. 
1546.    Doctrina  cristiana. 

1 546.  Cancionero  Spiritual.  The  first  book  to  bear  the  name  of  Juan  Pablos  as  the  printer, 
—  "Jua  pablos  Lobardo." 

1547.  Regla  Christiana  breue. 

0.  1 547.   Doctrina  cristiana  en  lengua  mexicana. 

1548.  Doctrina  Cristiana. 

1548.  Ordendqas  y  copilacion  de  leyes. 

1 548.  Doctrina  Cristiana  en  Lengua  Huasteca. 

1550.  Doctrina  cristiana. 

1 5  5  3  •  Doctrina  cristiana. 

1554.  Recognition  Svmmularum. 

1554.  Dialecta  resolutio  cum  textu  Aristotelis. 

1554.  Didlogos,  by  Cervantes  (Francisco  Salazar). 

1555.  Vn  vocabulario  en  la  lengua  Castellana y  Mexicana. 


In  155^  five  books  were  published,  among  them  the  Sumario  Compendioso. 
It  thus  appears  that  not  only  was  this  the  first  book  on  mathematics,  but  it  was  the 
first  textbook  of  any  kind,  except  for  religious  instruction,  to  be  published  outside 
of  Europe. 

In  his  Bibliografia  Mexicana  del  Sigh  XVI,  Icazbalceta  speaks  of  a  copy  in 
the  library  of  the  Convento  de  la  Merced  and  of  one  in  the  Ramirez  sale.  There  is 
also  one  in  the  Biblioteca  Nacional  at  Madrid,  from  which  three  folios  are  missing, 
and  it  is  this  copy  that  has  been  used  in  the  preparation  of  the  present  work,  the 
missing  portion  containing  parts  of  tables  not  included  in  this  edition.  There  is 
also  a  copy  in  the  British  Museum. 

The  author  of  the  Sumario  was  Juan  Diez,  a  native  of  the  Spanish  province  of 
Galicia,  a  companion  of  Cortes  in  the  conquest  of  New  Spain,  and  the  editor  of 
the  works  of  Juan  de  Avila,  known  as  "the  apostle  of  Andalusia,"  and  of  the 
Itinerario  of  the  Spanish  fleet  to  Yucatan  in  1 5 18.  He  is  sometimes  confused  with 
Juan  Diaz,  a  contemporary  theologian  and  author.  In  a  letter  written  to  Charles  V 
in  1533  he  is  mentioned  as  a  "clerigo  anciano  y  honrado,"  so  that  he  must  have 
been  advanced  in  years  when  the  Sumario  appeared.  That  this  was  the  case  is 
also  apparent  from  a  record  of  the  expedition  of  15 18  in  which  it  is  stated  that 
"triximus  vn  clerigo  que  dezia  joan  diaz,"  doubtless  a  young  and  adventurous 
apostle,  full  of  zeal  and  desire  to  make  known  the  gospel  in  the  New  World. 

The  other  four  books  appearing  from  this  press  in  the  year  1556  are  as  follows  : 

Costituciones  del  Arzobispado  y  Provincia  de  la  muy  insigne  y  mny  leal  ciudad  de 
Tentixtitlati  Mexico  de  la  Nueva  Espana; 

Costitutiones  Fratruum  Heremitarum  Sancti patris  nostri  Augustini Hiponensis  Episcopi 
et  doctoris  Ecclesiae  ; 

speculum  Conjugiorum ; 

Catecistno  y  Doctrina  Cristiana  en  Idiotna  Utlateco. 

Not  again  in  the  sixteenth  century  did  the  Mexican  printers  publish  any  work 
on  mathematics,  except  for  a  brief  Instruccion  Nautica  which  appeared  in  1587. 
The  press  was  generally  true  to  its  early  purpose  to  issue  only  books  relating  to 
the  conversion  of  the  native  inhabitants  to  the  way  of  the  Cross. 


Ill 

GENERAL  DESCRIPTION  OF  THE  BOOK 

The  Sumario  Compendioso  consists  of  one  hundred  and  three  foHos,  generally 
numbered.  After  the  dedication  (folios  i,  v,  and  ij,  r)  there  is  an  elaborate  set 
of  tables,  including  those  relating  to  the  purchase  price  of  various  grades  of  silver 
(folio  iij,  v),  to  per  cents  (folio  xlix,  r),  to  the  purchase  price  of  gold  (folio  Ivij,  v), 
to  assays  (folio  [Ixxxj,  r\),  and  to  monetary  affairs  of  various  kinds. 

The  mathematical  text  (folio  xcj,  v)  consists  of  twenty-four  pages  besides  the 
colophon  (folio  ciij,  v).  Of  these  pages,  eighteen  relate  chiefly  to  arithmetic  and 
six  to  algebra. 

The  signatures  are  a  (j,  •  •  •,  iiij  [  •  •  •,  viij]),  and  similarly  for  b,  •  •  .,  i,  k,  1, 
m,  n(j,  ...,iiij  [,  •••,vij]). 

Folio  i,  r  consists  of  the  arms  as  shown  in  the  facsimile,  and  the  title : 
(H,  Sumario  copedioso  delas  quetas/de  plata  y  oro  q  en  los  reynos  del  Piru  son 
necessarias  a  los  mercaderes  :  y  todo  genero  de  tratantes.  Co  algunas  reglas  tocan- 
tes  al  Arithmetica.    Fecho  por  Juan  Diez  freyle. 

Folios  i,  V  and  ij,  r  contain  the  dedication  to  the  viceroy,  beginning : 
<II,A1   Illustrissimo   Senor  Don   Luys  /  de  Velasco  Visorrey  y  gouernador  d'la 
nueua  Espana.  /  &c.    Juan  diez  freyle :  que  perpetua  felicidad  le  dessea. 

As  to  himself  the  author  says : 

"  Por  quato  Jua  diez  freyle  estate  pre  /  sente  enesta  ciudad  de  Mexico  me  a 
becho  relacio  ql  co  ci  a  cu  y  /  dado  trabajo  &  industria  a  copuesto  vn  libro  de  quetas 
de  plata  &  /  oro  co  algunas  reglas  t'  uera  del  ordinario :  tocates  al  arismetica :  el 
ql  es  de  /  mucha  vtilidad  &  puecho  pa  en  los  reynos  del  Piru  a  causa  d'las  muchas  / 
variedades  q  enel  ay  enlas  leyes  de  plata  &  oro  &  otras  cosas  q  alia  le  vsa  /  lasqles 
todas  estan  en  el  dicho  libro  muy  copiosamete  puestas." 

He  therefore  undertook  the  work  for  the  purpose  of  assisting  those  who  were 
engaged  in  the  buying  of  the  gold  and  silver  which  was  already  being  taken  from 
the  mines  of  Peru  and  Mexico  for  the  further  enriching  of  the  moneyed  class  and 
the  rulers  of  Spain.  The  author  felt  that  he  could  best  serve  this  purpose  by 
preparing  such  a  set  of  tables  as  should  relieve  these  merchants  as  far  as  possible 
from  any  necessity  for  computation.  For  this  he  had  very  good  precedent,  not  so 
much  in  Spain  as  in  Italy.  In  1503  Anton  Bartholomeo  di  Paxi  had  published  in 
Venice  a  Tariffa  de  pexi  e  mesvre  containing  numerous  tables  relating  to  weight, 
value,  and  the  like,  and  intended  for  the  Venetian  merchants  engaged  in  foreign 
trade ;  in  1535  Giovanni  Mariani  had  published  a  Tariffa  perpetva  in  the  same 
city  and  intended  for  a  similar  purpose  among  the  merchants  of  all  of  Northern 

7 


Italy ;  and  besides  these,  various  other  works  of  a  similar  nature  had  already  been 
issued  with  the  intention  of  relieving  merchants  from  the  extensive  calculations 
imposed  upon  them  by  the  complex  systems  of  measures  then  in  use. 

Apparently  prompted  by  the  further  demand  for  a  brief  treatment  of  arithmetic 
which  should  be  suited  to  the  needs  of  apprentices  in  the  counting  houses  of  the 
New  World,  the  author  devotes  eighteen  pages  to  the  subject  of  computation  and 
presents  it  in  a  manner  not  unworthy  of  the  European  writers  of  the  period. 

The  most  interesting  feature  of  the  work,  however,  is  neither  the  tables  nor 
the  arithmetic ;  it  consists  of  six  pages  devoted  to  algebra,  chiefly  relating  to  the 
quadratic  equation. 

The  reason  for  this  interest  will  be  appreciated  the  more  when  we  consider 
the  state  of  algebra  in  Europe  in  the  middle  of  the  sixteenth  century.  Puzzle 
problems  involving  numbers,  such  as  would  now  be  solved  by  algebra,  were  known 
to  the  Egyptians  in  the  second  millennium  e.g.  ;  but  no  treatise  upon  the  theory 
of  equations  is  known  before  about  a.d.  275,  when  Diophantus  wrote  his  great 
work.  It  is  not  until  the  beginning  of  the  ninth  century  that  the  word  algebra 
appears  in  its  present  sense,  having  first  been  used  by  al-Khowarizmi  in  a  treatise 
written  in  Bagdad  in  the  time  of  the  caliphs. 

In  the  Middle  Ages  there  appeared  a  number  of  algebraists  of  ability,  notably 
Leonardo  Fibonacci  of  Pisa,  who  lived  early  in  the  thirteenth  century ;  and  little 
by  little  these  scholars  added  to  the  store  of  material  which  had  already  accumu- 
lated in  the  works  of  the  later  Greeks  and  the  Orientals. 

With  the  advent  of  printing  from  movable  types,  in  the  second  half  of  the 
fifteenth  century,  there  was  awakened  a  new  interest  in  mathematics,  and  par- 
ticularly in  the  field  of  algebra.  The  Greek  and  oriental  writers  had  solved  the 
quadratic  equation,  but  the  equations  of  the  third  and  fourth  degrees  still  awaited 
solution,  and  a  better  symbolism  was  in  urgent  demand. 

The  middle  of  the  sixteenth  century  saw  the  solution  of  the  cubic  and 
biquadratic  equations  by  the  Italian  algebraists,  and  saw  numerous  efforts  made 
at  devising  a  convenient  symbolism. 

It  was  at  this  time  that  Juan  Diez  wrote.  There  had  already  appeared  the 
notable  algebra  of  Cardan  (the  Ars  Magna  of  1545),  the  Germans  had  pub- 
lished two  treatises  of  merit,  and  there  had  appeared  in  15 14,  from  the  pen  of 
Gillis  Vander  Hoecke,  a  Dutch  mathematician,  a  work  of  some  consequence ; 
but  the  number  of  treatises  printed  before  1556  was  small,  and  these  were  far 
from  being  popular.  It  is  therefore  of  considerable  interest  to  know  that  an 
obscure  writer  in  Mexico  should  have  produced  even  six  pages  on  the  subject 
at  this  early  period  in  the  development  of  printed  scientific  literature. 

8 


IV 

NATURE  OF  THE  TABLES 

The  general  nature  of  the  tables  may  be  seen  from  the  facsimile  on  page  lo. 
The  abbreviations  used  are  as  follows : 

ps  is  used  for  peso  and  pesos,  originally  a  certain  weight  of  metal,  like  pound, 
libra,  and  lira.  The  word  comes  from  the  Latin  pensum,  from  pendere,  "to  hang." 
From  the  same  root  we  have  such  words  as  poise,  which  also  appears  in  avoir- 
dupois, and  such  physical  terms  as  pendant  and  pendulum.  The  Castilian  libra^ 
which  found  its  way  into  Mexico,  was  about  1.014  avoirdupois  pounds. 

t  is  used  for  tomin  and  tontines.  The  tomin  was  the  eighth  part  of  a  peso, 
and  this  was  the  same  as  the  later  real.  The  peso  was  therefore  a  "piece  of 
eight."  The  tomin  was  also  5  of  a  peso  of  weight,  or  i  of  a  drachm.  The  name 
comes  from  the  Arabic  tomn,  "an  eighth  part." 

mros  is  used  for  maravedi  and  maravedis,  a  word  derived  from  the  name  of 
the  Moorish  dynasty,  Murdbitln,  during  which  the  coin  was  first  struck.  In  these 
tables  56  maravedis  make  i  tomin,  but  in  tables  of  a  later  period  the  real  {tomin) 
is  given  as  equivalent  to  34  maravedis  de  plata  Mexicanos.  In  some  of  the  tables 
of  Juan  Diez  the  tomin  is  taken  as  56^  maravedis  (fol.  Ivij),  and  there  are  several 
other  slight  variations  of  this  kind  in  the  tabular  work.  The  maravedi  is  also  used 
as  a  weight,  as  on  page  10. 

on  is  used  for  onqa,  our  ounce,  from  the  Latin  tmcia,  "a  twelfth  part,"  the  ounce 
being  the  twelfth  part  of  a  Roman  and  early  Spanish  pound.  From  the  same  root 
we  have  our  inch,  the  twelfth  part  of  a  foot.  The  uncial  script  of  the  Middle  Ages 
received  its  name  from  the  same  source. 

gfos  is  used  for  grano  and  granos,  our  grain  as  a  unit  of  weight,  one  twelfth 
of  a  tomin  ;  —  "  12.  granos  q  tien  vn  tomin." 

U  is  often  used  for  1000.  Thus,  we  have  ijU  for  2000,  iijU  for  3000,  and  so 
on.  The  name  for  U  is  cuento,  given  in  the  tables  as  ciieto,  a.  word  derived  from 
contar,  "to  reckon."  This  use  of  U  was  common  in  Spain  in  the  sixteenth  century 
and  has  an  interesting  history.  The  symbol  may  be  seen  in  the  last  two  lines  of 
the  facsimile  on  page  10,  where  300  maravedis  correspond  to  jU  pesos,  that  is, 
to  1000  pesos.  The  U  in  this  sense  is  of  uncertain  origin.  It  appears  a  century 
earlier  as  I J  and  may  possibly  have  come  from  one  of  the  several  Roman  symbols 
for  a  thousand.  Among  the  curious  variants  are  D  with  the  vertical  bar  dupli- 
cated, and  a  symbol  resembling  the  late  Greek  character  for  900.  In  the  sixteenth 
century  the  Portuguese  used  for  the  same  purpose  a  symbol,  the  cifrao,  which 
somewhat  resembled  our  present  dollar  sign. 

9 


I^imt^c.   m\lt>.  x>c\ct  fo»    iij 


media  otf 

Jon 

tioti 

iif«oii 

ih{.oii 

V.Oll 
Vl.OfI 

Vij.on 

UnfoQ 

i).mFo9 

ilimfod 

iiij.mfos 

v.mfoa 

vi.mroa 

vij.mfos 

vlif  mfo0 


P0  iS.trvi^.'  iij 
/  pB  I)  n 

i).p0iii).t* 
ij.p&vii.tirvHj.    U). 


iij  P0  Ij.tVWil  li 
vj pe  v.tjcvilj.  tl). 
r.  p0   t. 

Fiij.ps  i7,t.rv>:vl)/  ij 
pvj.po  v.t.rvilj.   lii» 

mu'.ue  i|,t.rFr\ni.  ij 


I*  niFos 
rnnfoo 

il-  in  foe 
1^1  iuiros 
rlijmroa 
cliij  mfo0 
clinjmrod 


lu.  v(.^0  v»t.|:viif, 
c.      pa   t. 
cmm^  if.trrrvjl 
ci:rjcvj60  V.t):PUN 
cvK     p0 

cvllij.ps  ij.t.r):>^vll, 
cilvj.^e  v.t.|:vii)t 
d.     p0 


it 
«)• 

.«. 
ij 

iii. 


rit)|.mfo0 
xlvUmfoe 
rtviijiT^foj 
rQji;.mr*o0 
I.  ,  mfp0 
I  Imfo0 
I  ij.mfoa 
I  iijjnro0 
Mj.  mf  00 


lv.mfo0 
Ivl.mfoe 
lvu.mfo0 
IvilJmfo0 
I  ij:.mfo0 
Vc.  mfo0 
Ijc  f.mfo0 
Irtj.mfoa 
iFUjjnfoa 


Iiriiij  mro0 
li:v.  mfo0 
Ui;.  mfo0 
UTjtinfo0 

j:c.  Ill  foe 
cm  foe 

cc  mf  00 
cccjnfoo  I 
ccccmf  00  j 


dlff.60 
clvj.^0 

cli:i|f.p0 

C|Kt)(.|0 

cljrF.  p0 
cUjLiij  p0 
d)L'|CVj.p6 
cUm'p0 


i|.tm-viii|. 

ii.tiftivlMf, 

ij  t.nrrvli  ij* 
V.t.jiVll).ilj» 


dm'H(.p0  ij.t.ririivli.ij. 
cli:ri:T|.p0  v.t.):viii.iij. 

^I».    p0 

crcifi.f0  ii  twirrll.fi 

ccili^0  irt.»i:vlMf. 
cci>)f0  vtpvilMif' 
ccir.  pa 


ccrfijp0 

cci:t>)«p0 

<a'):|ii).p0 

ccUvj.p0 

CCC.       p0 

Kccrw]  P0 
od]cv|.f0 

uccc^rllipa 


ilt.prrviili* 
vt.rvli).  il). 

V.t.j:vliJ  it|. 
v.t.jcp.iii.il> 

y.fmT«.ii 


The  approximate  nature  of  the  tables  may  be  seen  from  the  page  here  shown 
in  facsimile.  In  the  third  line  the  value  of  i  ounce  is  given  as  o  pesos  3  tomines 
i8|  maravedis.  The  half  ounce  should  then  be  worth  half  of  this,  or  i  tomin 
37|  maravedis,  as  stated.  The  quarter  ounce  should  then  be  worth  half  this  amount, 
or  46 1  maravedis,  whereas  it  is  given  in  the  table  as  only  46  maravedis.  Similar 
instances  of  a  lack  of  exactness  are  found  throughout  the  tables,  —  a  fact  that  would 
hardly  have  been  considered  significant  in  the  somewhat  crude  financial  transactions 
of  the  period. 

The  Roman  numerals  are  used  in  all  the  tables,  as  was  the  custom  among 
many  bankers  in  various  parts  of  Europe  until  the  close  of  the  seventeenth  century. 
Where  the  chief  commercial  and  financial  operations  consisted  in  additions  and 
subtractions,  these  numerals  were  nearly  as  convenient  for  purposes  of  practical 
computation  as  the  Hindu-Arabic  symbols  in  use  to-day. 

The  tables  extend  to  "  dos  mil.  cccc.  de  ley."  There  is  a  table  of  per  cents 
extending  to  30%.    In  this  there  are  such  entries  as 

iij .  por .  ciento.         c  .  ps         iij.  ps 

that  is,  3%  of  100  pesos  is  3  pesos. 

In  general  it  may  be  said  that  the  tables  give  the  value  of  various  numbers  of 
ounces  of  silver  in  pesos,  tomines,  and  maravedis. 

The  terms  />esos,  tomines,  maravedis,  and  varas  seem  more  acceptable  in  the 
translated  text  than  any  English  words,  and  hence  have  been  used.  The  more 
familiar  marks,  grains,  ounces,  crowns,  and  ducats  have  been  given  in  English. 

The  tables  are  no  longer  of  any  importance,  but  as  a  matter  of  interest  a 
single  page  is  here  shown  in  facsimile.  Only  the  mathematical  text  has  any 
historic  significance,  and  it  is  this  that  appears  in  the  translation. 


II 


THE  TEXT 

WITH  TRANSLATION  AND  NOTES 


gl  que  baftantemcntc  teitgo  pucfto  po2  nonde 
fm  (?a3er  qucrita  fe piicda  faber  el  Palo?  o c  quaU 
quicr  varraotejd"  ^c  plata  o  020  po2  oifcreiuc 
Ic^tpefoquc  tcnga  y^I^^Ioj  oelos  i^ntcrefcs 
quefe  acollumb^anaDar  por  qualqutcr  plata 
0  020  t>afla  trer  nta  po?  cCento.]^  4(Ti  mifino  cl\>a 
I02  DC  qualcfquier  pcfos  pc  plata  cozricnte  coni== 
pzados  DC  cnfji^ado  ra3onando  cl  ]rntcrc8  oc  ocl^©  a  vcf  ntc  po2  ci 
entoiuntamcntccontodoIoniasneccfanoDelanueiia  ^fpnna  co 
laa  rcdiicioncs  oc  pefos  oucados  i  co2ona8,De  aqui  adelantc  pon^ 
drc  algunad  reglae  oelas  nccefarias  en  los  rcf no8  V€\p€vn  (unta^ 
mente  con  algutiasquifhones  para  curiofos  enrrela/qt^s  van 
algunae  dcI  arte  ma^^oi  referuadae  al  algebra:  las  quates  conlo  oe^ 
maaflnofucrctalcomoconuieiterecebidla  volunmd  tfea  carita^ 
tiiiamente  cmendado^  lafalta  que  tuuiere. 


*&^^«a^??.^ 


Common    Rules 

"^OW  that  I  have  sufficiently  explained,*  without  doing  the  actual 
computing,  how  the  value  can  be  found  of  any  ingot  or  bar  of  silver 
or  gold  of  whatever  standard  or  weight,  and  how  to  find  the  amount 
of  commission  up  to  thirty  per  cent  which  it  is  customary  to  give  for 
any  gold  or  silver ;  and,  in  the  same  way,  how  to  ascertain  the  value  of  divers 
weights  of  silver  currency  bought  as  assayed,  reckoning  the  commission  from  eight 
to  twenty  per  cent,  together  with  all  else  that  is  necessary  in  regard  to  the  reduction 
of  pesos,  ducats,  and  crowns  in  New  Spain,  —  from  here  on,  I  shall  set  forth  some 
of  the  necessary  rules  which  are  used  in  the  kingdom  of  Peru,  together  with  certain 
problems  for  those  who  are  interested,  among  which  are  certain  parts  of  the  arte 
mayor  ^  pertaining  to  algebra.  If  these  with  the  rest  do  not  entirely  meet  with 
the  approval  of  the  reader,  may  he  accept  my  good  intentions,  and,  in  as  kindly 
a  spirit  as  possible,  excuse  the  mistakes  which  I  may  have  made. 

*  In  the  tables,  which  make  up  the  greater  part  of  the  book. 

t  The  arte  tnayor  was  a  term  commonly  used  in  the  sixteenth  century  for  algebra.  It  appears 
in  the  Latin  of  the  period  as  ars  magna  and  in  the  Italian  as  T  arte  maggiore.  Cardan,  for  example, 
called  his  great  work  on  algebra  by  the  name  of  Ars  Magna,  the  work  appearing  at  Niirnberg 
only  eleven  years  before  the  Sumario  Cotnpendioso  was  published. 

The  name  was  occasionally  combined  with  the  ancient  title,  "The  Science  of  Dark  Things," 
used  by  Ahmes,  an  Egyptian  mathematician  of  c.  1 550  B.C.,  and  with  the  Arabic  title  al-jabr  w'al 
muqdbalah,  used  by  al-Khowarizmi,  c.  820.  An  illustration  of  this  is  seen  in  the  title  of  Gosselin's 
treatise,  De  Arte  Magna,  seu  de  occulta  parte  numerorum  quae  et  Algebra  et  Almucabala  vulgo 
dicitur,  libri  IV,  which  appeared  in  Paris  in  1577. 

The  use  of  /'  arte  maggiore  for  higher  arithmetic  (algebra)  as  distinguished  from  /'  arte  minore 
for  elementary  arithmetic  may  have  been  suggested  by  the  seven  arti  maggiori  and  the  fourteen 
arti  minori  of  the  merchants  of  medieval  Florence. 

The  Italians  also  called  the  science  by  the  name  Regola  de  la  cosa,  the  reason  being  that  the 
unknown  quantity  was  called  the  cosa,  as  stated  on  page  51.  Because  of  this  fact  the  German 
algebraist  Rudolff  (1525)  called  his  treatise  Die  Coss,  and  English  writers  of  the  same  century 
spoke  of  algebra  as  the  "cossike  arte." 

The  Arabic  title  given  above  means  "restoration  and  equation,"  and  hence  algebra  came 
also  to  mean  "restoration  to  health."  It  is  for  this  reason  that,  in  Dofi  Quixote,  they  sent  for 
"  un  algebrista  who  attended  to  the  luckless  Samson." 


15 


CCapmilop:imeropojdqriairct>aa 

cmendcr  la  regla  para  t)a3<roc  pitta  comcntccnfaaida. 

Fltcndido  tengo  que  pocaj  rescs  fera  ncccfTario  t>a 
yr  quenta  que  pflHe  oelos  ve^nte poj cicnto  que ef 
ta  cfcripto:  pero  ocjcado  eflo  a  parte  Dare  aqui  la  re^ 
gla  para  que  con  faber  partir  la  ^aga  quien  quiera:]^ 
C8  que  ala  cantidad  q  quicres  faber  quanto  es  De  en 
faradoXana  diras  adclante  x)oS  3ero8  o  cifrascomo 
eftaSjOOji^DcrpuceaiiinaconcienroelinterefreqDaspo^loenra* 
f  ado,  po  ?  lo  qua  I  parte  aquello  a  q  anadilic  las  dos  ctfrad,  i  lo  que 
falierciiUpiirticionreranlodpcrosenrafadosa  q  fcbuelueloco?*^ 
riente.y  nota  q  lo  que  fob^areenla  parttcion  Ton  pefoe,^  q  loe  ]pzs 
vc  inultiplicar  po2.8.tomincsque  tiene  vn  pefo  t  lo  p2odu3ido  (pas 
i>e  partir  po?  el  partidoi  oc  antes  y.  cl  aduemniiento  fera  tomU 
nes'.r  anfi  mefmo  fi  algo  fobjare  foh  tomines  r  Ipas  Ics  vc  inultipli 
carpo:.  i2.grano3qtienevntcmintpartirlospo2clinefmo  par*? 
tidor.  I  claducnini lento  fera  granos  los  qualespon  conlospe*^ 
Tos  F  tomines  dc  las  particiones  r)eantC8\  y  aquello  fera  lo  que 
valclaplata  co^rientebueltaenenfa^ada,  y  feateauiroqucficn*» 
lo  co2rientc  oufere  tomines  que  po2  cada  vn  torain  podras  enlugar 
oelasDoscifras.u.^medio  tpojque  me)02lo  emiendaspondre 
aqulvnc^rcniplo. 

If  |g)Hmero  jE;:einpIo. 

^®iga  que  to  tengo,  4^iu  ps.  6.  tomines  oe plata  co:rienteloff 
quaks  quiero  c6p?ar  trc  plata  enfaii^ada  o  x>c  0:0  que  melo  ra  a.14 
po:  cicnto  Devntcrefc  para  loqunltengowtH?  que  ala  c5tidad  co 
rrknte  t)a8  Deanadir  Dos  cifra6  f  fi  ouicre  tomines  po2  cada  vno 
tn  lugar  oelas  cifras.i 2  .v  tnedlo  pojq  las  cifras  po2  fi  no  vale  nada 

m   iUj 


Common    Rules 

(I[^ Chapter  I,  in  which  is  explained  the  rule  for  finding  the 
value  of  assayed  silver. 


UNDERSTAND  that  sometimes  it  will  be  necessary  to  make  calculations 
above  the  prescribed  twenty  per  cent ;  but  aside  from  this  I  shall  now  give  a 
rule  which  anyone  who  knows  division  can  follow,  namely :  to  the  amount  of 
money  with  which  you  wish  to  buy  the  assayed  silver  annex  two  zeros  or 
ciphers  (oo) ;  then  compute  on  a  basis  of  a  hundred  the  commission  which  you 
give  for  the  assaying,  and  divide  by  this  the  number  with  the  two  ciphers  annexed  ; 
the  result  of  the  division  will  be  the  assayed  pesos  which  the  currency  will  buy. 
It  should  be  noticed  that  the  remainder  left  from  the  division  represents  pesos 
and  must  be  multiplied  by  8,  the  number  of  tomines  in  a  peso ;  this  product  must 
be  divided  by  the  same  divisor  as  before,  the  quotient  being  the  number  of  tomines. 
In  the  same  way  if  there  is  again  a  remainder,  it  represents  tomines  and  must  be 
multiplied  by  12,  the  number  of  grains  in  a  tomin,  and  if  we  divide  this  by  the 
same  divisor,  the  quotient  will  be  the  number  of  grains.  Now  put  these  with  the 
pesos  and  tomines  and  the  result  will  be  the  value  of  silver  currency  in  assayed 
form.  Let  me  also  say  that  if  you  wish  the  currency  in  tomines,  put  in  place  of 
the  ciphers  12  and  a  half  for  each  tomin,  and  in  order  that  you  may  better 
understand  I  give  an  example.* 

dL  First  example 

(2, Suppose  that  I  have  4321  pesos  6  tomines  in  silver  currency  with  which  I  wish 
to  buy  as  much  assayed  silver  or  gold  as  they  will  give  me  at  24  per  cent  com- 
mission.! Now  to  the  amount  of  currency  you  must  annex  two  ciphers ;  and  if 
you  wish  tomines,  for  each  one  put  in  place  of  the  ciphers  12  and  a  half, 
because  the  ciphers  themselves  are  not  of  any  value  when  considered  alone, 

*  The  author  here  makes  an  approach  to  the  decimal  fraction.  A  tomin  is  ^  of  a  peso,  and 
hence  4321  pesos  6  tomines  is  equal  to  4321  pesos  plus  6  x  0.12 J  pesos.  It  follows  that 
4321  pesos  6  tomines  is  equal  to  (432,100  +  6  x  12^)  hundredths  of  a  peso.  If,  now,  we  wish  to 
divide  4321  pesos  6  tomines  by  1.24,  we  may  avoid  decimal  fractions  by  dividing  432,175  by  124. 
This  is  what  the  author  does  in  the  illustrative  problem  which  follows. 

t  The  word  ynteres  (interest)  is  used  by  the  author  to  mean  any  kind  of  percentage. 


17 


tpucdasalH^jelatcrirucnPcattmetarentalmancra^IastJeatrat 
que  al  v>iio  i^i^cn  valer  cicnto  i  anlTi  ponicndo  el  y^\oi  oc  t>ii  to* 
iliin  0  ;)08  o  qualeo  quier  tomines  fimcn  po:  ft  t  po2  las  Dos  cifras 
pojQuanto  tiencn  cos  grgoos  que  fon  vnidadi:  uesenait  anm 
wermoaumemaalviioqrea.ioo.y  notaquecftpnocs  otracofa 
que  multiplicar  patioo.  7  que  fe  pone  aft  poj  mas  b2cuedad:pu^ 
cst02nandoanue(lroc;:mploravc^8que loco2riente es,  4321^ 
pefos.6«  tomines  pislpsqualcsponadclantc  en  lugaroelascifras 
75.qucs  fu  ipalo:  T)elos  feis  tomines  a  D03e  i  medio  cada  vno  t  vie 
nenafer.  432175.  losquales parte  po2  ciento f  ve^nte  ^quatro 
quefonclvalo:  pelos^ioo^relKntcrefer  venirte  aalaparticion. 
348s.peros.  rrobwn.35.losquales|?a5  tominesXqueesmulti^ 
pli  cando  loo  poj^S.r  Ton.  280.  que  partidospoj,  12  4,  vienen,2  .to. 
I  fob2a,32.lo8  quales  \^^  granos  que  es  multipUcjando  los  poz.  1 2 
quetienevntominf  ron,354.  quepartidospo:  ciento]^  vtKnter 
quJtrote  vendran.3,  granos  que.  juntos  con  looe  mas  Ion. 
;485.ps,2.to.3.grano8.Yc{loe8enloqueliquidamcntc  feto^nan 
loa.  452i.pclos.6.tomine8t)eco2riente  conip^ados  Deenfa^ado 
0  De  o;o  a,24  .po:  cicnto  T  ft  quieres  ver  ft  es  vcrdad  anadc  les  fti 

fntcrcikalos.24.po?.ioo,quefon.836.pc(bs.3.tomine8.o^gros 
fventrteavcriftmo. 

432i.ps.6.tominc8.co2r{cntc       000 
432175*  particlon       121 

124,  partfdoi       206 

348s.p8:2.to.3,5ros.enfapdo.  06293 

836.ps.3ato.9.^os.tntcrere.  17055 

ccloenfarado.  432175  |3485.p«.2.to.3.gfcs. 

32lob2a      I  '^4444      3^^ob2a.    z 

pOM2.Sro8.     022  12  J2        p02.$,tO.     O42 

roii.3$4  384  !  J  iifoiv  280         2S0I2 

«i4  124 


Common  Rules 


but  are  annexed  merely  to  raise  the  number  in  such  a  way  that  one  becomes  a 
hundred ;  and  substituting  the  value  of  as  many  tomines  as  you  wish,  they  serve, 
for  themselves  and  for  the  two  ciphers,  to  advance  the  number  two  places,  through 
units  and  tens,  thus  raising  one  to  lOO.  Observe  also  that  this  is  nothing  more 
than  multiplying  by  lOO  and  is  stated  in  this  way  for  brevity.  Now,  using  our 
example,  we  observe  that  the  amount  is  4321  pesos  6  tomines.  In  place  of  the 
two  ciphers  to  be  affixed,  substitute  75,  the  value  of  six  tomines,  each  being 
twelve  and  a  half  hundredths,  and  the  result  is  432,175.  Divide  this  by  one 
hundred  and  twenty-four,  the  value  of  the  100  plus  the  commission,  and  the  result 
of  the  division  is  3485  pesos,  with  a  remainder  of  35.  This  remainder  is  reduced 
to  tomines  by  multiplying  by  8,  the  result  being  280.  If  we  divide  this  280  by 
124,  we  have  2  tomines  with  a  remainder  of  32.  Multiplying  this  32  by  12,  the 
number  of  grains  in  a  tomin,  we  have  384,  which  divided  by  one  hundred  and 
twenty-four  gives  3  grains.*  The  entire  result  now  is  3485  pesos  2  tomines  3  grains, 
the  amount  of  assayed  silver  or  gold  purchased  with  4321  pesos  6  tomines  of  cur- 
rency at  24  per  cent.  If  you  wish  to  prove  this  to  be  true,  add  the  commission  at 
24  per  100,  which  is  836  pesos  3  tomines  9  grains,  and  you  will  see  that  it  checks. 

4321  pesos  6  tomines  currency 
432175  dividend 

1 24  divisor 

3485  pesos  2  tomines  3  grains  assayed 
836  pesos  3  tomines  9  grains 
commission  for  assaying 
32  remainder       i 
by  12  grains  022 

are  384  384  |  3 

124 


000 
121 

206 
06293 
17055 

432175  I  3485  pesos  2  tomines  3  grains  t 
124444        35  remainder        3 
1222      by  8  tomines  042 

1 1      are  280  280  |  2 

124 


*The  result  should  be  33^,  but  the  fraction  is  rejected.  This  shows  again  how  difficult  it 
was  to  perform  such  operations  to  a  high  degree  of  precision  without  the  aid  of  decimal  fractions. 
Indeed,  the  use  of  such  denominations  as  pesos,  tomines,  and  maravedis  was  due  solely  to  the 
necessity  experienced  by  the  ancients  for  avoiding  fractions.  For  example,  6  tomines  is  merely  a 
substitute  for  |  of  a  peso,  or  0.75  of  a  peso. 

t  The  method  of  division  here  shown  is  about  the  last  stage  of  the  medieval  galley  method 
which  had  been  in  use  in  Europe  for  a  long  time.  By  that  method  the  figures  were  canceled  out 
as  soon  as  they  had  served  their  purpose.  Evidently,  however,  the  Mexican  press  had  no  canceled 
figures  in  their  fonts,  and  hence  they  do  not  appear  in  this  text. 

19 


Cf  i^cdaracioii  oela  rcgU  pafada* 

Cl^ota  qiie  l3  rcgla  pafada  €s  vcrifimamflcnte  U  rcgla  t)c  tres  y  q 
affi  como  elU  I'c  funda  po2  plata  fe  puedc  fundar  po2  otras  mucl)a8 
vias  que  bieii  podria  x>t^iT:  lo  tengo^ioco.botijas  x>c  vino  que  las 
looo.ron .  2  5.po?.ioe.nia^b2€sqijc  lasotras.iooo.coinp^un  mc» 
las  ted  .18  con  c^cUe  clpicae  ticpojla  mcdida  t)elaa  grandesjEe*^ 
innncli>cnquanta8feboluerflnl98acoc.ct)ica8medida8po2Uinic 
dull  oclao  grandes.  I^ara  elta  i  las  fcmcjantcs  as  vc  fundar la  re« 

gina»cll4nuncra»SLi25.t)ela8c|>icasreto2nanen.ico.t>ela5gra 
dc5  c.'i  que  fe  tomara.i  oocoelas  ct)icas:triu]tipUca  las.  i coc.po: 
loo.fcranjoocoo.qcslorninKocomovcesqucafiadir  fidclantc 
las  t>os  cifrastpucs  parte.i  ooooo.poz.i  25.]^  renir  te|?an.  8oo-^ 
en  tantas  fc  tosnaran  lasjcocbotijascl^icasniedidaspojlamc 
dida  Dclas  g  r9ndes:la  p:iieua  es  que  Ics  cclpes  fu  intcrcfle  a,  2  5,po: 
ICO.'?  vciiir  te (?3n,2oc.coino  vcU  ficrurado 

C/ooQ.boti)riSGrandes.25,p0:,ico» 

inasqiicioooo.cj^icas. 

Cirtiiliiplicacioij 

CintiltipHcado? 

Sllop2odn3ido  looooo         106000  |$oo 

®;ru  partidoz 

^aDijcuimicntc 

C^ntcrcs 

Ifteascr  t)c  pefos  t)ucado8 1  oc  oucados 
pefosmu^enb^cuc. 
CSi  quifieres  fabcr,tanto8  pefos  quatos  t)ucad08  fon/acd  el  qui 
to  oclos  pcfos  '^  fainalo  condloa  mcfmoe  ^cl  rcniamcnte  icra  10  4 
t^cfl'eas  Tuber* 


1000 

00 

100 

624 

lOOOOO 

1 00000 

ITS 

12555 

800 

122 

li'^o 

1 

Common  Rules 

([[^Explanation  of  the  former  rule 

d. Observe  that  the  preceding  rule  is  really  the  Rule  of  Three,*  and  in  the  same 
way  that  this  applies  to  problems  relating  to  silver  it  can  be  applied  to  as  many 
other  processes  or  kinds  of  problems  as  one  may  wish.  For  example,  I  have  2000 
jugs  of  wine,  1000  are  25  per  lOot  larger  than  the  other  1000.  Buy  them  all  from 
me  so  that  you  will  get  the  small  ones  at  a  price  proportional  to  the  price  of  the 
large  ones.  First  find  how  many  of  the  small  jugs  are  equivalent  to  1000  of  the 
large  jugs.  For  this  and  for  like  problems  the  rule  can  be  applied  in  this  way :  If 
125  of  the  small  ones  are  equal  to  100  of  the  large  ones,  to  how  many  large  ones 
are  the  1000  small  ones  equal  ?  Multiply  the  1000  by  100  and  the  result,  100,000, 
is  the  same  as  if  you  had  annexed  two  ciphers  ;  now  divide  by  1 2  5  and  the  quotient 
is  800,  the  equivalent  of  1000  small  jugs.  To  prove  this,  increase  the  number 
800  by  25  per  100  and  the  result  is  1000,  as  you  see  worked  out  below. 

(11,1 000  large  jugs,  25  per  100  more  than  1000  small  ones. 


1 800 


(H,  Multiplicand 

1000 

00 

Cn,  Multiplier 

100 

024 

(H,  Product 

lOOOOO 

I 00000 

CH,  Divisor 

125 

1255s 

CL  Quotient 

800 

122 

(U,  Percentage  t 

200 

I 

<n.A  short  method  of  changing  pesos  Into  ducats  and  ducats 

into  pesos 

(Q,  If  you  wish  to  find  how  many  ducats  there  are  in  a  certain  number  of  pesos,  take 
one  fifth  of  the  number  of  pesos,  add  it  to  the  number  of  pesos,  and  the  result  is 
what  you  wish  to  find. 

*  The  Rule  of  Three  was  the  most  popular  of  all  the  medieval  commercial  rules.  It  came  to 
Europe,  through  the  Arabs,  from  the  Hindu  arithmeticians.  In  the  Middle  Ages  it  went  by  such 
names  as  the  JRegula  de  Tribus,  Regula  Rerum  Trium,  Regula  Aurea,  and  Regula  Mercatorum. 

\  The  Latin  form  was  i^  per  centum,  whence  2$por  ciento,  25  pc,  25  p^,  253,  and  25%. 

t  It  will  be  observed  that  there  are  two  inaccuracies  in  the  original  edition,  namely,  loooo 
appears  for  1000,  and  280  for  200,  "ynteres." 


21 


^445,$i«  el  quinto  es.Sp.quc  (umtdo9  funtos  montan.$|4.]r  tan 

tosDucadod  ronlo9,445*pa. 

0i.26S,^9xl  quinto  c».53.ruinado8  colo8.265,  fon.;i9*cpmo  vets 

^Si  quificret  fabcr  vnt  cantidad  oc  t>ucado8  quantos  pcfos  fori 
fdcadfefinooe  lo8  oueadoa^Kloqucreilaieferaloquebufcas. 

CSM'^u^^^®  ^^  ^^^  ca.$9,r€(ladoo  oe,5;4.quedan.445.tti 

ro8  pcrodronlo8,5^4*9ucado0. 

49^3 1  S,9ucad08:el  fclmo  c8,53.re(lados  De«3 1  S.quedan.  2  64* 

41ledu3ir  pefod  a  maraaedCo  fin  m  ultiplicar. 
%Si  quificre8  fabcr  vm  onim  inu]^  biicua  pnra  faber  vna  catidad 
T>e  pefo8  quant08  marauedts  (on  po:  mui  inae  f  acil  f  b^cue  manc^ 
ra 4  inultiplicar:(;>a5 108 pcfodmillared  ffaca  el  ofesmo^?  t>elo que 
rellare  la  mitad  r,  aquello  fcra  loque  oeffcad  faber, 

IfEjcemplo. 
^)Coma.456,p8.ba5 108  mdlarea  rott.456ooc,  el  r)ic3mo  odoa 
qualc8  e8.4$6oo.C  como  ve^s  figiirado)  q  rertados  Del  p>incipal 
qtted«,4io4oo.lamitadc8.2o$2Qojnf8*  ^ tanto inontan  los 
456.pcl08a  ra3onoe,45o,marauedi8clpero> 
4$6.p8.  Ton       4  $  6  ooo« 

£1  x>icyno es        4  <;6oo. 
Ilellan  41  0400. 

3L  t  initadea,      20^2  oo.marauedi8> 
CSeracfante  ala  paffada  en  ma  eo.>  cantidad. 
f:Coma*9456S.pcfo8.4.roiiiinc0 1  b^slos  mdlarca  ^  porcl  me 
d(o  pefo  poii.5oo»i!rerfln  .34568soo.el ^icsmo  es.  54  5685o.rc* 
ftadoa  ocl  p:incipalqucdan*^ii  u^5o.la  mltad  e8.i5S5s825,mara 
ucdia  como  veno  pozla  figura. 


Common  Rules 

((L  Example 

CD, One  fifth  of  445  pesos  is  equal  to  89,  which  added  to  445   gives   534,  the 

number  of  ducats  in  445  pesos. 

dLOne  fifth  of  265  pesos  is  53.   Adding  this  to  265  gives  318,  as  you  see. 

(J,  If  you  wish  to  find  out  how  many  pesos  there  are  in  a  number  of  ducats,  subtract 
one  sixth  of  the  number  of  ducats  from  the  number  of  ducats. 

dL  Example 

<D[One  sixth    of    534    ducats    is   equal    to  89,    which    subtracted   from    534    is 

equal  to  445,  and  this  is  the  number  of  pesos  in  534  ducats. 

dLOne  sixth  of  318  ducats  is  equal  to  53,  which  subtracted  from  318  gives  265. 

([^To  reduce  pesos  to  maravedis  without  multiplying 

CLIf  you  wish  a  good  rule  for  finding  the  number  of  maravedis  in  a  certain 
number  of  pesos  by  a  much  easier  and  shorter  method  than  that  of  multiplying, 
raise  the  number  of  pesos  to  thousands,  find  one  tenth,  subtract  it  from  the 
number  of  thousands,  and  then  find  one  half  of  the  remainder ;  this  gives  the 
required  number. 

dL  Example 

<n, Raise  456  pesos  to  thousands  and  you  have  456,000;  one  tenth  of  this 
is  equal  to  45,600  (as  you  see  by  the  work);  this  subtracted  from  the  principal 
leaves  410,400,  the  half  of  which  is  205,200  maravedis,  and  this  is  the  number 
of  maravedis  in  456  pesos  at  the  rate  of  450  maravedis  to  a  peso. 


456  pesos  are       456000. 
The  tenth  is  45600. 


The  remainder    410400. 

The  half  is  205200.  maravedis. 

([[^Similar  to  the  above,  using  larger  figures 

CLTake  34,568  pesos  4  tomines  and  raise  them  to  thousands,  and  for  the 
half  peso  put  500,  and  the  result  will  be  34,568,500.  One  tenth  of  this  is 
3,456,850;  and  this  subtracted  from  the  principal  leaves  31,111,650.  One  half 
of  this  is  15,555,825  maravedis,  as  you  see  by  the  work. 

23 


^y 0 ella ttngopo^  iiie*    ^£1  oie5mo.        9456850,      ' 
jo2crcrb:euc^ciertaco    ^tKclljan,  31111050, 

mo  vcras  fi  lo  vflig.  1^2,  a  initaciesTTj  5  55  ^ismi*. 

IfSi  qui/icrco  faber  tmtoe  pcfoe  quant^ic  coronas  forrfin  }?st^tr* 
lopoa»i*irauccJi3,inuUipIica  lospcfoo  po:  nucuet  parte  po^fietc, 
t  el  aducniiiiicnto  feran  cc:ona8» 

^EpcinpIo» 
^56.ps.multiplu'^po?.9/oiU5o4.p.^rtepo:.7.v(ene.72,CQ2ona8. 

(Jlpani  leaser  iJUjcojonao  pefosji'iikiplica  poj  .7,  t  parte  po;.9 

^^riEFcmpIo. 
|56>co:onn3,ir;ultipl:capoz.7jfon.  44i.p:.rtepo:.9  vicnen.z|9 
<]uc  foil  pcfoo  po2 la fiirnaitaml  len  k  piicdc  t>a5er tjefta  niflncra.a*^ 
loa.56.perosafiade  fuc  Doefetcnes  que  fon.i  6.]^reran  laa 72,0020 
nao,la  orra  t5Ia9.63.co2ona8  rcfla  (us  too  noucnce  que  Ton.  14  .que 
dm  loo,49.pero8  €omo  vtie. 

^J^ara  leaser  T)e  Ducados  co2ona8 1  rnr  co^oms  t)ucado0,aunque 
cfla  piieflo  po2quenta,multiplica  I08  r'U^adoo  po2. 15 .  ^  parte  po: 
14. r  pa  t)n3er  T>cco2ona8  Ducados^fnultiplica  P02.14,  y  P^''^^^  V^^ 
qii|m3e,t  I08  vltimcs  aduenimientoe  feraii  lo  que  bufcae, 

€|;£;:cinplo. 
C42,t)ucadoJ,mulHolicapo?..i5.roii,63o,partepo2.i4,rvcnirtc 
I?an.45.]ftanta8|to2ona8fonlo8.42.oucado8. 

S£):einplo. 
, ,  ca  po2.i4,ron.S4o, parte po:.t5.\>(e]ae.56r 

]?'tanto8i>ucado8fonlae.6c,co2ona8:tambien  lopuedes  bajcrco 
mo  lo  palTado:]?:  ee  q«ealo8.42,r>ucadc8  ajufled  fu  cato25fluo,  que 
cetree;  con  que  fon  la8.45.co2ona8;t  afTi  mifmoalas    .  o.  cc2cna8 


Common  Rules 

34568  pesos  4  tomines  34568500. 

(n.1  consider  this  the  shortest  and  best      d^The  tenth        3456850. 


and  surest  method,  as  you  will  see  if  you      (Q^ Remainder     31 1 1 1650. 


use  it.  <n,The  half         15555825.  maravedis. 

<II.If  you  wish  to  find  how  many  crowns  there  are  in  a  certain  number  of  pesos, 
without  reducing  to  maravedis,  multiply  the  pesos  by  nine  and  divide  by  seven,  and 
the  quotient  will  be  crowns. 

(J^  Example 
(11,56  pesos  multiplied  by  9  is  504.    Divide  by  7  and  the  result  is  72  crowns. 
<II,To  reduce  crowns  to  pesos,  multiply  by  7  and  divide  by  9. 

dL  Example 

(II.63  crowns  multiplied  by  7  is  equal  to  441.  Divide  this  by  9  and  the  result  is  49, 
which  result  is  pesos.  These  two  rules  may  be  worked  out  in  this  way :  For  the 
first  rule,  to  56  pesos  add  its  two  sevenths,  which  is  16,  and  the  result  will  be  72 
crowns.  For  the  second  rule,  from  63  crowns  subtract  its  two  ninths,  which  is  14, 
and  the  remainder  is  49  pesos  as  you  see. 

(n,To  reduce  any  number  of  ducats  to  crowns,  multiply  the  ducats  by  1 5  and  divide 
by  14;  and  to  reduce  crowns  to  ducats  multiply  by  14  and  divide  by  15.  The 
quotients  will  be  the  desired  numbers. 

dL  Example 

CII.42  ducats  multiplied  by  15  is  equal  to  630;  divide  this  by  14  and  the  result  is 
45,  and  so  many  crowns  are  the  42  ducats. 

dL  Example 

(II,6o  crowns  multiplied  by  14  is  equal  to  840 ;  divide  this  by  15  and  the  result  is 
56,  and  so  many  ducats  are  the  60  crowns. 

Also  you  may  reduce  the  first  of  the  above  sums,  involving  the  42  ducats, 
in  this  way:  add  to  42  its  one  fourteenth,  which  is  three,  giving  45  crowns. 
Proceeding  in  a  similar  manner  with  the  second,  from  the  60  crowns,  since  they  are 

25 


poiquc  Viencnamcno0,rcilak8fuqum3auoquc  t8,4.qdcdl  (os 

Q;SuiTo9ememo2ia: 
f;S(  qulficrcs  faber  0€  mcm02U  muE  fa  cil  i;  verifllmamente  vna 
cantidad  oc  maraucdis  quantos  pcfoo  tbn,Dobla  lo9  mlllarcs  t  oef 
pued  anadeles  fu  oiesmo  t)afta  que  no  af  a  ce5enas,^  ela  fuma  poj 
Mdav>nididto:iu.^o.mar4ucdi8;|:poHinqo2  locrtticrMtaa  nota 
cftc  cf:emi)lo  po^que  inc  parccc  te  baftara. 

^£)cenipIo. 
ifComa.  1 2oooQ.!naraucdi8,oobla  los  miUarcs  Ton.  24  o.el  vley 
mo €8.24.^:00.24  ^o9^qiodo9(^on,266.^cfos,r  oclos rcF8«3oo, 
nmsiucdxByt  tamo  valcnlo«»i2oooo,marauedi8» 

^i^udb  para  faber  loqucfe  oeue  oc  quiiuo  oc  qaaW 
quier  pUu  cojricntc  que  fc  fucre  a  quiiitar. 

<[Si  fucrcs  a  quiiitar  alguna  plata  co^ricnte  y  quifierce  fiber  poj 
h  pluni.i  0  DC  cabc^.i  (oque  te  ban  oe  lleuarbc  quuito,toma  el  quar 
to  oelo  que  fucrcs  a  quintar  rasonando  el  marco  8,4. ps.?  ocfpuea 
toma  vno  poMoo.oclo  mefmo  y  fuinalo  concllo ,  y  la  fuma  fera  lo 
que  oeues  Del  quinto  f  oerecbos  oel  vno  po:  ciento*.]^  para  mas  U" 
tiffacion  tu^a  pondre  vna  figure, 

f:/6):cmplo. 
iI!Coma.4^75.p8  iz  Pnara]?apo2  0cbnp?luegofacad  quarto 
cd.i  14  ;.petb8.6.torTpned  r  oefpuee  pon  ocbajco  vno  po2  ciento  ^ 
loque  fuiileaquintaf  quefon,4^.pefos.6.tomme8  pozquanto  loT 
75.  que  te  fobjaron  foil  treo  quartos  oc.  1 00.  y  afli  los  tree  quar** 
to9  oe  T>n  pefo fon. 6. tominestlo  qual fuma  conlos  oe  mas  f  mon** 
.raraii.ii$9  pefo9«.4  toifiincs:?  tc!ntoedloquete(panoelleuaroc 
:quimo  toerecl^oscomo  ve|;8  Hgurado, 


Common  Rules  , 

to  be  reduced,  subtract  the  fifteenth  part,  which  is  4,  leaving  56  ducats. 

d^A  rule  to  be  memorized 

<n.If  you  wish  to  memorize  an  easy  and  sure  way  for  finding  the  number  of  pesos 
in  a  number  of  maravedis,  double  the  number  of  thousands,  then  to  the  result 
add  its  tenth,  and  so  on  until  there  are  no  more  tens.  Then  for  every  unit  in 
the  sum  take  50  maravedis.  In  order  that  you  may  better  understand  this  rule, 
consider  the  following  example  which  seems  to  me  to  be  sufficient. 

dL  Example 

(H^Take  120,000  maravedis,  double  the  thousands,  and  the  result  is  240 ;  the  tenth 
of  this  number  is  24,  and  the  tenth  of  24  is  two,  and  the  sum  total  is  266  pesos. 
In  units  column  there  is  6,  and  for  each  of  these  units  take  50  maravedis  and 
the  result  is  300  maravedis.  Thus  we  have  266  pesos  300  maravedis,  which  is 
the  value  of  120,000  maravedis. 

(JA  rule  for  finding  the  tax  on  any  amount 
of  silver  currency  to  be  assessed 

CII.If  you  are  to  compute  the  tax  on  any  amount  of  silver  currency  and  wish  to 
know  how  much  they  will  demand,  take  one  fourth  of  the  amount  to  be  assessed, 
reckoning  the  mark  at  4  pesos  ;  then  take  one  per  cent  of  the  original  amount,  add 
the  one  fourth  and  one  per  100,  and  the  sum  will  be  the  tax  and  the  one  per  cent 
fee  demanded.   For  a  better  understanding  I  shall  set  forth  an  example. 

dL  Example 

<n.Take  4575  pesos,  draw  a  line  underneath,  below  this  write  one  fourth  of  the 
number,  or  1 143  pesos  6  tomines,  and  below  this  write  one  per  cent  of  what  you 
took  to  be  assessed,  or  45  pesos  6  tomines.  The  75  which  is  left  over  is  three  fourths 
of  100,  and  the  three  fourths  of  a  peso  is  6  tomines.  Add  these  together  and  the 
sum,  1 1 89  pesos  4  tomines,  is  what  they  will  demand  of  your  money,  including 
the  fee,  as  you  will  see  in  the  work  below.* 

*Thatis,  25%+i%=i+ iJ^. 


27 


q  f\  \o  |?i3icrc0  poj  mar     %£X  qrtOr    j  i4^,pei08^6^<C 

co«,quct)a8iDctoinar     ipor,ioo,   45pPeioi,^, 

;pd2cadamarco,i3]^9,r      ^Son,        u89,pe(bs,4.tor 
^pojcada  on^a  Til,  to,|^ 

po2  cada  quarta  tree  gnmo8,]r  mac  d  Tno  pordcmo'Coinatf go  ^f 
ct)©,^  nota  que  todo  lo  ucl  vno  po:  cicntoes  lo5  T)ctcc^)o«  oel  iwar 
cado?,r  no  !na5>po2q  aunq  a  d  Ic  viene  d  qrto  mas  De  i>erccj|>a  que 
a^  fc  Ic  oa  (u  inagel^ad  lo  lleua  dc  mcnos:  t  la  caufa  es  que  le  pagt 
el  vrro  po2  cicnto  Delo  que  le  ricne  x>c  quinto  como  tu  odo  qlleual^ 
a  qiiinrar  o  po2  nie|'o?tie3ir  fe  lo  qutnra  dc  balde* 
^ilf^uf  mucl)a8manera8a!r  DemuliiplicarcntrelaSquttlcsTOtf 
goeltapo2lame|o2]f  Dcinasvcrdad  ?  certiduir.b^e  lopnopojno 
fe  lleuar  n  Jda  i5  memo2ia  lo  otro  pojque  para  la  regla  x>t  tree  no  ea 
nccefTario  mudar  Us  letris  para  aucr  oe  multipUcar. 

4[.i£|reniplo. 
tlBiQO  que  multipI!quea,879,po2,75S.lo  qual  Ihij  t)e.ponerenla 
manera  como  vea  figurado  i  oar  cntre  la  mulnplicacion  i  muliipll 
cado:  vna  vara  como  efla.V  I  oar  ;)ebapoot*ra  raf  a  coiiiaaquivc* 
n.87S\Q7S,r  luego  c6el8,p2imera  Ictra  oe  mano  ^3quferdiqocf 
mutripUcadormtdtipUcatodaslaaoda  multiplicaddnquefoiilita 
oe  addanteoeliaraM  pontendb  laaletraaendla  manera,S„Te3ea,9 
ron,7:^,d  fieteqtfceaoe3ena  Debajrood,8,i:d,2,quee5i>fi(dadTii 
grado  adelame  que  f  rra  oebajco  i)d,7,r  fino  ou  tera  vnfdad  auiaa  tf 
poner  la  oe3ena  en  Ik  mcfmo  lugarocba^rcodaletra  quetomafte 
po2multiplicado2f  poj  lavnidadend  gradoaddantevnseroco^ 
mo  efle.o.t  fino  oulera  oe5enana auiaa  oe  poner  nad«*4)eroli  wf 
dad  en  fu  lugnr  vn  grad<yadelante  oeoo  aula  ;^eff  aria  i>c3ciia  t  lot 
gDOl»8,x>e3ca,7.5C),d,s,ocba)rood,2,rd,6,Tn0radoadcl«teoel 
2,rluegODi.8,ve3ej,8,64,el,6,pebappelotro,6,Fd,4,Tngrado 
eddimtc,q]ucea  Pcbap  pclap2imcraIctr4T)cteiniritipItoaon;«gp> 


4575  pesos 

<II.The  fourth 

1 143  pesos  6 

tomines 

I  per  lOO 

45  pesos  6 

tomines 

(H.  Makes 

1 1 89  pesos  4 

tomines 

Common   Rules 

en.  Let  me  advise  that  if  you  wish  to  do 

this  by  marks  of  weight,  for  each  mark 

take  I  peso ;   for  each  ounce,  one  tomin ; 

for  each  quarta,  three  grains,  and  besides 

the  one  per  cent  of  which  I  have  spoken. 

Observe  that  the  entire  one  per  cent  is  the  fee  for  the  weigher  or  assayer,  and 

no  more,  because,  although  there  is  due  him  one  fourth  more  as  a  fee  than  is 

given  him  by  his  majesty,  he  receives  less  than  this.    The  reason  is  that  he  pays 

to  his  majesty  one  per  cent  of  that  which  is  due  himself  of  the  tax,  as  you  do  on 

what  you  take  to  be  taxed ;  or  better  expressed,  it  is  taxed  gratis. 

dl, There  are  many  ways  of  multiplying,*  among  which  I  consider  the  following 

to  be  best  and  most  accurate.     For  one  reason,  no  memory  work  is  required ; 

and  for  another,  according  to  the  Rule  of  Three  it  is  not  necessary  to  move  the 

figures  in  order  to  know  how  to  multiply. 


(J^  Example 

<II.To  multiply  879  by  758,  we  must  place  the  figures  in  order  you  see  set  forth, 
and  draw  between  multiplicand  and  multiplier  a  line  like  this\,  and  then  draw 
underneath  them  another  line,  thus:  875X978.  Then,  using  the  figure  8,  the 
first  one  on  the  left-hand  side,  which  is  the  multiplier,  multiply  all  the  figures 
in  the  multiplicand  which  are  beyond  the  line  in  the  following  manner:  8  times 
9  is  72  ;  place  the  7,  which  is  tens,  under  8,  and  the  2,  which  is  units,  under 
the  next  figure,  which  is  7.  If  you  have  no  units,  you  must  put  the  tens  in  their 
place  under  the  first  figure  which  you  take  as  the  multiplier,  and  in  the  place  of 
units  you  must  write  a  zero,  like  this :  o.  If  you  have  no  tens,  you  must  not 
put  down  anything,  but  must  put  the  units  in  the  column  next  to  the  one  in 
which  there  would  have  been  tens.  Then  observe  that  8  times  the  7  of  the  multi- 
plicand is  56,  and  so  we  place  the  5  under  2,  and  place  6  in  the  next  column 
after  2.  Similarly,  8  times  8  is  64,  and  we  place  the  6  under  the  other  6  and 
place  4  in  the  next  column  under  the  first  figure  of  the  multiplicand. 

*  The  method  most  commonly  used  in  the  sixteenth  century  was  not  the  one  given  in  the 
example,  but  one  of  the  two  known  by  the  Italian  names  gelosia  and  bericuocolo. 


29 


f«Det4d,8,lftomocl,7,veocllteconclqaab{,7>«€jeJnifeuc63 
4,6/pebap  oel,5  ,cnla  mcfma  pidcn  oc  el,7iTr  cl  tree  vii  grado  ade 
lliue  ^^fojK\,6,i  liM:g6"b<jr,vc5ca  fictc,  49,etq»^JltT0  ocb  jp 
itl,3,t,<l,9.^cbiij:o  ocl,4^Pea4clantcluoga  oi,7,vcy.f  ,3,56,cUi 

dilrtraocla multipl(caoi6:^^»oc|:»cl,7,^to    gT5\97g 
n>ad,5,qucc8portrcro  muUipl(cado:concl ql    7^^  4^^ 
pi,5,vc5ca.9,45,cl4f^cbai:o  ocl otro,4,cnU       $6  9^ 
o«dcn,?Jl,5.quctom«ftepoirnultiplicado2fcl,       oj  54 
Sr^Aaco 0:1,5, tJ  oclantcieUicgoi>i,5,vc5cd^7        4  5 

35/l,j'.i>cbapDcl,5,tcl,5,x)cba|:ot>d,6,y luc^ 4  ?. 

(5ioi,5,V(e5C«,8,4o.cl,4,Dcba):oDcl.5,T cl 3e'^    ^  5S  7SQ    , 
raoebaKQ  ^<:U  poftrcraletra  i  lucgo  fuiiia. 

i[!1l\cgU  para  cobiar  oc  fu  ina(;cl  I  ad  po:  c!  qumto. 
C'Ahjc|poS<;reo  af  oa  i^u^oqle8X)cuciu  nat^ciiadoincrooi^pa 
loS  aucr  oe  cobiar  Ics  C6  necefl'ar  io  buicar  0:0  0  pkt«i  po^  quirar  pa 
raque  po2  d  quinro  (p  cot^zen  loa  qualcG  po;  no  faber  Io  que  an  t^  lie 
uari quiiitar para cobur Io q fc Ico T)cuc  0 llcuaii  ocinas 0 oe  n^cv 
no9  oelo  qual  lleiimdo  oc  mais  Ico  c$  per)ur3io  en  que  la  plata  )^  0:0 
qucUtuan  lea  quella  intereQ'c  1^  lleudndo  oe  mcnos  lea  cs  nefcciVa^ 
ryo  bolucr  otu  vc}  a  quintar  ^  qpe  tengo  entcndido  algucoo  rccU 
bciip;radumb;ef  paralodtjdeocQAaEudax^emodoarc  aqui  vna 
rtf  la  coino  facil  ifm  ningun  fcirro  tepan  Io  que  \fA  oc  llcuar  a  quiii 
mrparacob^aral  )u(loloqiiele6Dcuc. 

f:ileg1a* 
Cltenloquefctcoeucpo^ri^nia  aU  qualafiade  oos  seroaeomo 
(ii:p$;.oo.|;  d  oulerc  tomitiea  pon  pot  cada  tomin  en  ^g^r  oeloe  5e 
Qoa  005c  t  medio.conio  cengo  X}{c\^  enla  rcgla  oe^o^ijuu'pUta  co2' 
rientea  enfafada  t  oefpuca  parte  aquella  cicidad^^  vcf^ate  7  (cfS 
I  iQ  que  ala  partfdon  falicrc  fera  Io  que  ocflfeas  faber  como  ^\vti^ 


726 

460 

56 

95 

63 

54 

4 

5 

4 

3 

855 

750 

Common  Rules 

Now  leave  the  8  and  take  7  for  the  multipher,  thus  :  7  times  9  is  63,  so  we  place  the 

6  under  5  in  the  same  column  as  the  7,  and  place  the  3  in  the  next  column  under 

the  6.    Then  7  times  7  is  49,  and  we  place  the  4  under  the  3,  and  the  9  under 

the  4  in  the  column  of  the  first  figure  of  the  multiplicand.  Then  7  times 

8  is  56,  and  we  place  the  5  under  the  9,  and  the  6  under  the  7  in  the     875X978 

column  of  the  second  figure  of  the  multiplicand.    Now  leave  the  7  and 

take  5,  the  last  figure  of  the  multiplier.    Then  5  times  9  is  45,  and 

we  place  the  4  under  the  other  4  in  the  column  of  5,  the  multiplier, 

and  put  the  5  under  the  other  5.   Then  5  times  7  is  35,  and  we  place 

the  3  under  the  5,  and  the  5  under  the  6.   Then  5  times  8  is  40, 

and  we  place  the  4  under  the  5,  and  the  zero  under  the  last  figure  of 

the  multiplicand,  and  then  we  add. 

((L  Rule  for  collecting  what  is  due  from  his  majesty 

(0,1  believe  there  are  and  have  been  many  to  whom  his  majesty  owes  money.  For 
those  who  have  to  collect,  it  is  necessary  for  them  to  bring  gold  or  silver  to  be 
assessed  so  that  they  may  collect  what  is  due.  These  people,  because  they  do  not 
know  how  much  they  have  to  take  to  be  assessed  in  order  to  collect  what  is  due 
them,  take  too  much  or  too  little.  Taking  too  much  is  hurtful  because  commission 
is  charged  on  the  extra  gold  or  silver  that  they  take.  Taking  too  little  makes  it 
necessary  to  assess  again,  from  which,  I  understand,  some  unpleasantness  results. 
For  such  as  they,  with  the  help  of  God,  I  will  state  here  an  easy  rule,  without  any 
error,  by  which  they  may  know  how  much  to  take  to  be  assessed  so  as  to  collect 
justly  what  is  due  them. 

CLRule 

<n,Put  down  the  sum  that  is  due  you  ;  to  it  annex  two  zeros  like  these  :  00.  If  you 
would  like  tomines,  put  for  each  tomin  twelve  and  a  half  in  place  of  the  zeros, 
as  I  have  stated  in  the  rule  for  converting  currency  into  assayed  silver.  Divide 
the  sum  by  twenty-six,  and  the  quotient  will  be  what  you  wish  to  know,  as  you 
will  now  see. 


31 


^Sfgo  que  te  Dcuan,i]44,pfrod  1 3' 

alos  qualcs  anade  I08  <oo8  3cro8 1  0300 

monta^ra,!  14400  lo  qual  ptepoz  i ,  440614400 

26,t  Ticnc  ala  piirtiaon,44oo:ps  2  666  6 

quca,4,pcro8cl  marcofon,  iioo,  221 

marco8rtanto8(?a8^1lcuaraqui    fi'^juews,   l.^oo,lnfo8^ 
tarparacob2arlo8,ii44,pcro8la    Cson.      _4^4oop# 
pJUcuaferaquefaqucJel-VrDcrpii    Cfclqfto    MxISISt 
C8 vno po2 cicnto  como te tcngo oi    yi,po.Moo,      44^^* 
ct)o cnla  fegimda  rcgla antc>  pclla.    4®^"*     ^TT^Zpaf" 
iE5c  aqui  adclate  pondrc  algunae  p2Cgunta8  que  arniqucHoJonW 
ccflarias  para  lo  que  cncftc rei^no  fc  vfa  \  I08  que  fon  igficionados 
ala  qucntti  fc  (polgaran  con  clla8  po^qucaunque  nofonfutile^jiini 
I08  que  algo  faben  I08  que  lo  oedean  Taler  con  ellaa  tendran  pJiM 
pio8paraina8rubir4 

Q1|i^2imer«'t»?egunta» 
€[!^ui'aqumtar  derto  oibnofelo  que  paguet>eIqu(nropo;qine 
Ioquitaronr)CPnalibMn(a,perorequc  facadoe  quint08  ^9ere« 
c|p08  me  codo  efle  020  (in  tntcrelTe^i  5  S  49pero8:ocmando  quanf 
to  C8  lo  que  me  quiraron  enla  Ub:an(a  y.  que  te  \o  que  agoia  valee^ 
lie  o:o.1\egIa  faca  el  -^  t>e,i  5  8  4,  ed,^  9  6,  ru«i«Io8  €0,1584^0 
1 9  80J08  qualc8  parte  poa,99,el  aduenimicnto  cs  vcfnte  afsfhi^ 
do8  a/98o,fon,2ooo,t  wnto  vale  ago^a  elo^o/uma^jp^/i^iio 
fonj4i  6;,i;  tanto  fe  quito  rcla  Ub:an(^a  perilfima 

|{;S?gnnda  p^egunta  * 
^Semcfante  ala  paffada  fui  a  quintar  vn  pcdaijo  oe  0:0  no  fe  To  5 
pefaua  antes  0  quintar  pero  fe  que  me  quitaron  ^el,4 1 6,  pefoa  dc 
Derccl?o5,  lEJemado  quejlo  que  me  (?a  De  qoedanlRegla  multiplier 
4i6,po2,5/on,  2c8o,  partelo»po2,26,irienen,8o/c(lado8«e,. 
2080,  quedan^  2000,  pcloaqualca  faca  el-fe8^oo,rcftado© 


CX) 

12 

0300 

1 14400 1 4400 

26666 

222 

(H.  Proof 

1 100  marks 

CLAre 

4400  pesos 

CLThe  fourth    i  lOO  pesos 

(11,1  per  lOO 

1         44  pesos 

(n.They  are 

1 144  pesos 

Common  Rules 

dL  Example 

<n.  Suppose  that  there  is  due  you  1 144  pesos.  Annex 
to  this  number  two  zeros,  making  1 14400  ;  divide  this 
number  by  26  and  the  result  is  4400  pesos  which,  at 
4  pesos  to  the  mark,  make  11 00  marks,  the  amount 
you  ought  to  take  to  collect  the  1 144  pesos.  To  prove 
this,  find  \  of  4400,  and  then  one  per  cent  of  4400, 
as  I  have  stated  in  the  second  rule  preceding  this  one, 
and  then  add. 

From  now  on  I  shall  propose  certain  questions 
in  which,  although  not  necessary  for  what  is  used  in 
this  kingdom,  those  that  like  arithmetic  will  delight. 

Although  they  are  not  difficult  for  those  who  know  something  of  mathematics,  those 
who  desire  to  know  more  will  find  in  them  the  beginnings  for  further  advance. 

(J,  First  problem 
(11,1  took  a  certain  amount  of  gold  to  have  taxed.    I  do  not  know  what  tax  I  paid, 
because  it  was  paid  from  a  bill  of  exchange,  but  I  know  that,  deducting  the  tax 
and  the  fees,  this  gold  cost  me  without  commission  1 584  pesos.    I  want  to  know  how 
much  they  took  from  me  in  the  bill  of  exchange  and  what  the  gold  is  now  worth. 

Rule :  Take  |  of  1584,  which  is  396  ;  add  this  to  1584  and  the  result  is  1980. 
Now  dividing  1980  by  99  the  quotient  is  20,  and  this  added  to  1980  gives  2000,  the 
present  value  of  the  gold.  Adding  396  to  20  gives  416,  and  this  is  what  was  taken 
from  the  bill  of  exchange.* 

dL  Second  problem 
dissimilar  to  the  above,  I  went  to  assess  a  piece  of  gold.    I  do  not  know  what  it 
weighed  before  being  assessed  but  I  know  that  they  took  416  pesos  as  the  fee. 
I  want  to  know  what  should  be  left  for  me. 

Rule:  Multiplying  416  by  5  we  have  2080;  dividing  this  by  26  there  results 
80.    This  subtracted  from  2080  gives  2000,  from  which  find  |,  which  is  400 ; 

*We  have  25%  of  1584  =  396,  tax;  125%  of  1584  =  1980,  value  of  gold  less  1%.  Hence 
the  value  is  2000.   Then  2000  —  1980  =  20,  fee,  and  396  +  20  =  416,  total  payment. 


33 


niC0la6O5dfiiarf^0< 


i>c,i  ooo,quedan,i6oo>DcIo«  qu  I'V  &  f^iea  vno  goi  ciHo  fom  6  /c 
lUdos  6>i  6oo^qucda.is^4  ,y  tai  o  <8  lo  q  ago^  vale  d  020,JQb^s 
b^cuc  a]ufUoo8  3ero8  comoelloJ,oo,adcUite  oc,4i6/on,4iCoo 
partcpaj,26^iBTieri,i 6oo,X)clojjqualc5faca  wopo:,ioo,vUnJ 
i6/ffta9oo  Dc^iOoo  queda  n  i<  843P02  lafuma. 

CJCciccfap^eptita. 
^iijEcompiado  olcj  varaJDcierciopclo  mcno8,iOope(b8po?,J4, 
pefos  vmaTvna  \?ara  dtcrcfopclo,i5m5doacomoco[lolavaraw[\e 
gla  fuma  los  pero8,2o,ir  J  4  /on,  54,quc  fcra  tn  particion  rcfta  oc 
U8,io,vara8  la  \>i\a  que  0136 1  mas  ,qucdau,9  ,vara8  po2  la8  qua^ 
lc8  utc  lo5,S4  ,plcne,6,v:  tanto  cs  cl  pzecCo  dc  cada  vaia|p>2ucua,i  o 
vara8a,6,pcro8  |?a5ciir6o,pero8  meno8,2o,pcro8  qucdaii,4o,Dl 
5e  que  coftaron,34,pcro8 1  mas  vna  vara  que ron,6,con  que  valC 
los  mcrmo8,4o,  pefoo* 

l^duartap^egunta* 
IflE  comp^ado.i  i^varas  Delo  olct>o  menoaj^cpefos  po?,98,  ps 
menos  quatro  varas  T>emado  a  como  cofto  la  vara,lRotfl  ella  q  e« 
muijbicuc  t  veriflfimafumalos  pcro8,3o,r.98,ron,i28,rumalas 
varas,i 2  f  54,roii;i6,poilo8  quake  parte, 1 2  8,vcnir  te  an,8,t  ta^ 
toes  clpjedooe  cada  vara»1l^2ueua,i2,varaaa58,ron,96,mcno8 
30,  t9,66y  ^(3e  co(tar5 ,98,meno8,4,vara8  que  ron,3  ^^p^ros  que 
como  vei:s,quedanlosoicbo9> 66. 

lj;C^u(ntap2egu'nra. 
^g  comp:ado,9, varas  T)cloT)(cJ^o  po2  tanto  mas  oc,4o,peros, 
quanto,n,vara8  al  mcfmo  pxedo  valen  menos  oe.yo.Demado  a  co 
mo^oi^la  vara.Kegla  ^>a5como«la  pafladafuina  los  peros,4o  j 
7csilbii,i  io»  fum^  las '»>ar3^,9  ,f  ,1  ^/on  ,2  2  ,poHo3  quales  parte  loJ 
iio^  ftdwenimieiito  C8,^  ,yr  tato  es  cl  p2ccio  DC  cadttvara  .]^2ueua, 
9»vaNs  a,5,pdb8  ron,4^.pcrosqueron,5,masDc4o,Y,i3,varas 
a,5,fon.65  ,quc  fon^s.pcfos  menos  w,70jcomo  vers* 


Common  Rules 

this  subtracted  from  2000  leaves  1600,  of  which  find  one  per  cent,  which  is  16; 
and  this  subtracted  from  1600  gives  1584,  the  present  value  of  the  gold.  Briefly, 
annex  two  zeros  to  416,  making  41,600;  then  divide  by  26,  giving  1600.  Take 
one  per  100  of  this,  or  16,  and  1600  minus  this  gives  1584,  the  required  sum,* 

dL Third  problem 
<n,I  bought  10  varas  of  velvet  at  20  pesos  less  than  cost,  for  34  pesos  plus  a  vara 
of  velvet.    How  much  did  it  cost  a  vara  ? 

Rule :  Add  20  pesos  to  34  pesos,  making  54  pesos,  which  will  be  your  dividend. 
Subtract  one  from  10  varas,  leaving  9.    Divide  54  by  9,  giving  6,  the  price  per  vara. 

Proof:  10  varas  at  6  pesos  is  60  pesos.    This  minus  20  pesos  is  40.    You  paid 
34  pesos  plus  a  vara  costing  6  pesos,  and  this  gives  the  result,  40  pesos,  t 

(^  Fourth  problem 
<^\  bought  12  varas  of  velvet  at  30  pesos  less  than  cost,  for  98  pesos  minus 
4  varas.    How  much  was  the  cost  per  vara  1   The  following  is  a  short  method  : 
add  the  30  pesos  and  the  98  pesos,  making  128  ;  add  the  number  of  varas,  12  and 
4,  making  16  ;  divide  128  by  16,  giving  8,  the  price  per  vara. 

Proof:  1 2  varas  at  8  pesos  are  96  pesos  ;  this  less  30  pesos  is  66  pesos.    You 
paid  98  pesos  minus  4  varas,  or  32,  and  this  leaves  66.X 

([[^  Fifth  problem 
(11,1  bought  9  varas  of  velvet  for  as  much  more  than  40  pesos  as  13  varas  at  the 
same  price  is  less  than  70  pesos.    How  much  did  a  vara  cost  t 

Rule:  Add  the  pesos,  40  and  70,  making  no.    Add  the  varas,  9  and  13, 
making  22.    Dividing  1 10  by  22  the  quotient  is  5,  the  price  of  each  vara. 

Proof:  9  varas  at  5  pesos  are  45  pesos,  which  is  5  more  than  40  pesos;  and 
13  varas  at  5  pesos  are  65,  which  is  5  pesos  less  than  70,  as  you  see.  § 

*  The  second  method  amounts  to  this :  f  i  o  x  —  20  =  34  +  x, 
416  =  I  %  +  25  %  of  amount  assessed  jr  =  6. 

=  26%  of  amount  assessed.  1 1 2  r  —  30  =  98  —  4  ;r, 
Vy*-  X  416  =  1600,  amount  assessed.  .i-  =  8. 

I  %  of  1 600  =       1 6,  fee.  §    9  .r  —  40  =  70  —  1 3  jr, 
Therefore        1584  is  left  after  the  fee  is  paid.  x  =  5. 


35 


quatoa3*  varne  alo  mifmo,  fini  mcnos;        70      22 

oc«  70.^3.  *  10       cd 

I  io|5.§8 
^@ui(ltone8  po2  Io5numero8  qdrados        2  2 

Ci^raaucrT)c  baser  qualquicr  p2cguntaquctefucrcdniandada 
t)C  numcros  quadradoo^es  ncccfTario  que  fcpas  quecs  numcro  q* 
drado  I  po2quc  fc  llama  quadrado,^  que  co  numcro  cubo  fpo?quc 
fe  llama  cubo* 

Cl^umcros  quadrados  fcllanian  t  fon  aqudloJqucnaccn  «la  mul 
tiplicacion  o  fon  pwdusidos  oc  algun  numcro  en  otro  femcfamc  co 
maAo,i6,7c4d,4,nacci)el>2,multipUcadcpo2rimermooC3icn 
do,2Tve3e8  2M^,t  el,9.nace  T>el,5,po2elmcrmo  connguiente 
po2quc  j,ve3C3,j,ron,9.  ocloo  quales  numaoolos  lincalca  como 
<l.2,oct3,ronla8raH5C8i 

•riRumcros  cubicos  fc llaman  r  fon  aqueltos  que  fon  contenidoS  d 
Siumero9tguafolinealc8Delaqualmult(plicacjo;ifonr»ic|?o^^^^ 

«20creado8amcomo,8.27,64,iJ^,^c.po2quc,8,na«^^^^^ 

puc8  end  pjoduydo  ca  a  faber,  2,  ve5e8,2,4>i:.  2,ve3e«,4>'^".  ^> 

fc3>^c3e8,3,9,w*^3c«>9y^7-'^c* 

f^rimcraquilHon* 


mcvntatmmeroquebamftandole.i5,bjig«^ 
d'rado  ?rcftando  t)el,4/ea  lo  mefmo.regla  <«»"^»  ^1^ '4»p";\?;^^ 


Square  Numbers 


13 

40 

9 

70 

22 

10 

00 

iio|5  pesos 

22 

(11,9  varas  for  as  much  more  than  40  pesos 
as  1 3  varas  at  the  same  price  is  less  than 
70  pesos.* 


dL  Problems  relating  to  square  numbers 

CD.  In  order  to  know  how  to  solve  any  problem  that  is  given  to  you  relating  to 
square  numbers  it  is  necessary  to  know  what  a  square  number  is  and  why  it  is 
called  a  square,  and  also  what  a  cube  number  is  and  why  it  is  called  a  cube. 

(II.A  square  number  is  a  number  that  is  derived  by  the  multiplication  of  a 
number  by  itself,  as  is  the  case  with  4,  9,  16,  &c.  The  4  comes  from  multiply- 
ing 2  by  itself,  as  when  we  say  that  2  times  2  is  4  ;  and  the  9  is  the  product 
of  3  multiplied  by  itself,  because  3  times  3  is  9.  Of  such  numbers  the  lineals 
like  the  2  or  the  3  are  called  the  roots. t 

CII,A  cube  number  is  a  number  that  contains  the  three  identical  numbers  multiplied 
together,  as  is  the  case  with  8,  27,  64,  125,  &c. ;  for  8  is  the  product  of  2  times 
2  times  2  ;  similarly,  3  times  3  times  3  are  27,  &c. 

([I^  First  problem 

dLGive  me  a  number  which,  increased  by  15,  is  a  square  number;  and  decreased 
by  4  is  also  a  square  number,  t 

Rule  for  solving :  Add  1 5  and  4,  making  19 ;  then  add  i  to  this  result,  making  20. 
Now  take  the  half  of  this  number  20,  which  is  10,  and  then  square  this  result, 

*  The  work  here  given  shows  the  cumbersome  method  used  in  solving  the  equation 

9  jjr  —  40  =  70  —  1 3  X. 

t  In  this  work  the  word  root,  taken  by  itself,  signifies  square  root. 

(a  Jf  b  ^  l\* 
j  —  a  —  b  (or  +  a)  is  a  square. 


37 


ei\Mon€e*V^t\03mmace. 

ao,io,vc5e8,!0,ron,i  oOyVchs  quak^  rcfla  los^i  5,qwc  fe  aii  x>c  a^u 
ftar  quecUft,S5,^  cilc  C8  cl  numcro  Demand  ido ,  oc  et  quol  <i  rcftrs 
loJ,4,qda,Si,fu  raH3cs.9.'raffi  mcfiiio  fi  Ic  a|uitac;to.as,fon  cicto 
rai^3  ^^  5^^^  ^^>*  o^poiqoc,!  cvcsce^i  o/on  cutQ  ij  tflo  vaila, 

^Scgundaqiiiftion. 

^3i)amcvnnumcroqucafu(landole,S/caquadr$idoYrc(tHndo^I 
S,qucdcquadrado,tomamcdioococt>oc8,4,quadralfe8^i'65a)u 
ftale,i,C8,i7,l?  cite  es  clnumcro  T>cmandando,  ^t  qual  (t  a|u(la5,5, 
t«3e,2  5^uc  fu  raY3  c«,5,t  fi  Ic  rcfta058,qucdan^9  ,quc  ftiTa^5e5j 
po:qttc,3,ve5eo,3,lon  nuciic  conio  TC][a. 

iflEn  imio2  cantidad^T^ame  vn  numero  que  afuft^ddle^io'/ca  m 
mcf 9  quadrado  |[  reftanio  9el,2o,quf  dc  quadrado.fcma  thitad  ^ 
2o,c«,io,  quadra  C8,ioo,0fu(la  vno  l?a3c»ioi,t  efte  esd  vn  numc 
ro  4  fi  lca)ufUa^20jd  quadrado  ?  fi  le  refia8,2o>qucdaquadi^o« 

^^uiftiontercera. 

CCicnevnoD08rei(a8mutbttena8t«nTcpo2enad.8.|^8>'npb^ 
quU:reDar,x>icnevnoc6p»fela5poivara9CHeflamanera  que  leva 
poacadaTara  t>ecada  tiki  tamos  tomincs  quantodvarafftutnere 
a^uelta  pic^a  oidas  i  t>icc|>a  la  quimtaho  |;>aUan  que  v^nitm^nfk 
lo8.S.pef<>8  que  t)iua  el  pnifim,oemado  que  varas  tenia  oada  vna 
po?  fi  (ura  b  quah  egla  es  (inM9jW(ter  que  Inifque^  i>oe  nmncro^  ^ 
drafip^ rdes,que  )unto5  en  vnono fean  ni as  que  Tno  i|dcro 
t>agaii  miinao quadrado  loo  qbaUo  nomeros  Um;}^jL  ,flucmtit 
liplicadoocada  wopo2  fi  meriiool5i(cndo,3»vc3ef,|tt.4  vc5ej,4 
i6,t  Jjimadoo  el  pbo  co  el  otrofon.i5»y  fti  ra5ei^,vloc  jitJi  pot 
reglit>c,5»ficin5ora^3oe,25/onvenido5  0cJj^qttce8clval()20c» 


Questions  relating  to  numbers 

thus  :  lo  times  lo  is  loo.  From  this  subtract  15,  and  we  have  85,  and  this  is  the 
number  required,  that  is,  the  one  from  which  if  you  subtract  4  you  have  81, 
the  root  of  which  is  9.  The  same  thing  happens  if  you  add  the  15,  the  result 
being  a  hundred,  the  root  of  which  is  10 ;  for  10  times  10  is  100,  which  checks. 

dL  Second  problem 

(H,  Required  a  number  which  increased  by  8  is  a  square,  and  decreased  by  8  is  also 
a  square.  Take  half  of  eight,  which  is  4  ;  square  it,  making  16  ;  add  i,  making  17, 
and  this  is  the  number  which  increased  by  8  is  25,  the  root  of  which  is  5  ;  and 
which  decreased  by  8  is  9,  the  root  of  which  is  3  ;  for  3  times  3  is  9,  as  you  see.* 

(Housing  larger  numbers,  required  a  number  which  increased  by  20  is  a  square 
number,  and  decreased  by  20  is  also  a  square  number.  Take  half  of  20,  namely 
10;  square  it,  making  100;  add  one,  making  loi,  and  this  is  a  number  which 
increased  by  20  is  a  square,  and  decreased  by  20  is  also  a  square. 

dL  Third  problem 

<II.A  man  has  two  very  good  ropes  for  which  he  can  get  8  pesos,  but  he  refuses 
the  offer.  Someone  offers  to  buy  them  by  the  vara  in  such  a  way  that  for  every 
vara  in  each  rope  he  gets  as  many  tomines  as  there  are  varas  in  that  rope.  When 
the  computation  is  made,  they  find  that  the  money  is  no  more  than  the  8  pesos 
which  the  first  one  offered.    How  many  varas  are  there  in  each  rope  ? 

To  solve  this  it  is  necessary  to  find  two  numbers  whose  squares  added  together 
make  a  number  which  is  no  greater  than  that  of  the  one  given.  These  numbers 
are  3  and  4.  Multiply  each  number  by  itself  and  we  have  3  times  3  which  is  9, 
and  4  times  4  which  is  16  ;  and  these  added  together  are  25,  and  the  root  of  this 
is  5.   Then  by  the  Rule  of  Three,  as  five,  the  square  root  of  25,  is  to  8,  the  value 


♦  This  depends  on  the  fact  that 


—  +  I  +  x-=  [  — - —  I  »  a  square, 


x^                    Ix  —  z\^ 
and  V  I  —  x=  ( 1  »  also  a  square, 

x^ 
the  rule  simply  giving  the  expression  —  +  I. 

4 


39 


lo^ft  vcndio^cado  vcndran  j^r,4,quc  fiicronlosnimierostana 
do8multiplica,8>po2,3,  ron,24,  parte pojcincopicncnqiwtroy 
4- 1  ^'t«^  ^on  Lis  Viiras  X)cU  vrw,luc^o  ^iyS^vc^B^^Jfon^^Mr 
^cpot,s,^i<^ncn,6,  -r Tcftasfonlas vaMsDclaotraqucfumadad 
cnifanibasticnen,! I, vanish  vn  quimo'?pcnd(da8cadavn«pot 
fi  tSdo  po2  cada  vara  t>c  cadaa^tanto5  tomincc  quitas  raroG  tiene 
vfentn  a  valcr  loo,8,ps,la  p2ucua,nmltiplica,6-~4>02,6,-|-itoicn 

4o,r^f'nu^t»pl»ca,4.t.4»q»»^ntoJpo^4»T-f,vfen?,ij,t.jHl 
fiim;»iio8ron,64,tominc8  /^partido8  po2>S/omin€6queti€hcvn 

pcfofoii.S.pcfoa^ 

9€tttiftionquarc8» 

€^Si  cc  fif  cffe  pedfda  tnia  qtrfflfon  en  tal  manera  que  te  9l):eflen,T)a 
tne  vn  nuincro  quadradi^i;  tal  qu^  quirand  o  oel  vna  cantidad  cier 
ra  qiicde  quadrado  f  tjitOmdoiiU  lea  quadrado,  para  auer  oeab*^ 
(bluer  vna  tal  quiftion  csticceflbrto  que  fepae  que  coTaceDttmcro 
congruo  i  que  cofaea^nimero  con^ruentc 

i|:iRaniero  cogruo  ft  Ilaffla  t  ed  vn  tal  numero  ques  abto  a  tar  trc 
cebir  otro  nuiiieroel  qual  fe  llama  conGniaue  en  tal  manera  que  vH 
dole  0  rcCi  Wendole  fiemp^c  fta  quadrado,r  para  que  nicfojtinaff 
cUramcte  toentiend^s  ijondreaquibajrotosniitncros  conges  ^ 
coiigruentcdquemepareceTifran  netenartoa,ranil  niefmo  podrc 
Tncj:emplo,po!  cl  q\  (l  bienltnotas  podra©  oeclarar  tcdaslas  quif 
tfonco  q  po  >  ef>a  via  te  ftie  xn  i>eniandadao  ftedo  tal  el  mtmcro  tje* 
madado  q  Te  t)alle  numero  cognicnte  q  partloo  porel  eladuenlml^ 
to fea nuincro quadrado, po^no  to  ficndoa jotrofccrcto, dqual 
pef  0  po2  nofcr  p^oUco* 


Square  Numbers 


at  which  it  was  sold,  so  we  have  3  and  4  to  the  numbers  to  be  found.  Multiply  8 
by  3  and  we  have  24  ;  divide  by  five  and  we  have  four  and  |,  and  this  is  the  number 
of  varas  in  one  piece.  Then  8  times  4  is  32  ;  divide  by  5  and  we  have  6,  |,  and 
this  is  the  number  of  varas  in  the  other  piece.  These  added  together  give  1 1  varas 
and  a  fifth.  If  we  pay  for  each  vara  of  each  rope  as  many  tomines  as  there  are 
varas,  the  value  comes  to  8  pesos. 

Proof:  Multiply  6|  by  6|  and  we  have  40  and  ||.  Multiply  4  and  4  fifths  by 
4  and  ^  and  we  have  23  and  -^^,  which  added  together  gives  64  tomines.  This 
divided  by  8,  the  tomines  in  a  peso,  gives  8  pesos.* 

([^Fourth  problem 

<n, Suppose  that  you  were  given  this  problem:  Find  a  square  number  such  that  if 
we  take  from  it  a  certain  number,  there  remains  a  square ;  and  if  we  add  to  it  the 
same  number  it  is  also  a  square.  In  order  to  solve  such  a  problem  it  is  necessary 
to  know  the  nature  of  a  congruous  number  and  of  a  congruent  number. 

<D[A  congruous  number  is  such  a  square  number  that,  subtracting  from  or  adding 
to  it  another  number,  called  a  congruent  number,  it  will  still  be  a  square.  So  that 
you  may  better  and  more  clearly  understand  I  set  forth  below  the  congruous  and 
congruent  numbers  which  I  think  necessary,  and  I  also  give  an  example  by  which, 
through  careful  examination,  you  will  be  able  to  solve  all  the  problems  of  this  kind 
that  may  be  proposed.  The  number  required  must  be  such  that  when  the  congruent 
is  divided  by  it  the  quotient  will  be  a  square ;  if  it  is  not,  there  is  another  secret 
way,  but  this  I  will  not  give  lest  I  be  too  prolix,  t 

*  We  have  x^  ■\-  y^  =  8^.  If  we  take  z/  =  3  and  w  =  4,  we  have  v^  ■\-  w"^  =  5^.  Hence  the 
author  assumes  that  8:5  =  jr :  3,  and  that  8:5  =  y.  \. 

t  This  congruent  number  is  what  Leonardo  Fibonacci  (1225)  called  a  congruum,  a  number  of 
the  form  ^ xy {x  +  y){x  —  y).  He  gives  the  problem:  "To  find  a  number  which,  being  added  to 
or  subtracted  from  a  square  number,  leaves  a  square  number,"  and  uses  the  identities 

{X^  +  j2)2  _  4  ^yf^^i  _  ^2)  ^^yij^2xy-  x'^f, 
(x^  +  y^f  +  4  xy{x^  -  j2)  =  (x^+  2xy-  y'^. 


41 


.25,1Re.r ^a*  24*    <i>^a  mc  vn  num^ro^rado tal 

lOcIKe t  Da^  ^6^    que a|uftadole^6i5J)agamifliero q* 

i^p.lRc..^ T)a«  120^    iiMo-t  rcflancfo 0€l^6.qucde nu^ 
.2i$»15lt\r  D^-  216.    mero  quadrado^parajloqu^il  t^ao 

aSj^K.'f.p  oa^  ^40*    Ocbufcar vnmlflU!nq;o  congrue 

40oJf\e.i?  pa^  ^84.!    tcquejpartkndole  po2,6,vega  im 
^is.Ke^oa*  3}6\jj,ooo*    meroquadradoci qualcomo a fue 

676,1Re.t  oa*  48o,.    ra  vchs  c!  pnicro  C8ji4,puc8  par 

841,'iRe.lJ  oa«  S40.    te^24,po2,6,vicne,4,quccs  qua 

900.1Re.^  oa.  So4*    dradoC^fu  ra^^  co  ooO?  luegoto 

us6»1Re.jH  oa.  960^    ma  cl  numero  coiigruo  quadrado 

ixiSJKc.i Pa»  1176*    co^refpodicntc ocde  nuraero con 

iii2.1Ke.i[ T)a^  toSo-    gruciite  que c8,25,partalcpoHo8 

i68i,lRc.]fT)a.  72o»    4,qucc8claducnimicnto;)cetj^ 

2025JI\c.]fDai  *944*    mcravicncn,6,  -^-^aqucftccs 
2soo.IRe.K0a.1j44  2400*    cl  numero  ocmudado  que  file a/u 

2^02.1l\e.|;  oa*  2160*    (las  fers  t>a3e5i  2 ,  ~-  que  e«  nu** 

2704.1Rc.r x>a»  i92o«.    mero quadrado  z fu  rarseSj^,-!- 

2So9.Il\c.KDa.  asio^    ^jfireftac  x)el,6,  qucda  -^tfa 

5O25.1Rc.\20a»  290 j*    ra£3C8  -^  po^quc  media  pc3  mc 

j364»7Rc.roa»  3360*    dlaes  -^E el mefmoce quadra* 

j6oo.ll\e.  F  oa.  345^^    ^^  ^li*<^  ft'  ra V3 e8  008 1  medigt 
322i.llle.ii^  r>a.  1320^  Jtain  ali/8^ 

4225.IRe.KOa,  2Q26» 


Questions  Relating  to  Numbers 

([L  Congruous       ([^Congruents  dL  Example 

(H^Find  a  square  number  which  being  increased 
by  6  will  still  be  a  square,  and  which  being 
decreased  by  6  will  also  be  a  square.  To  solve, 
you  must  find  a  congruent  number  which  being 
divided  by  6  the  quotient  will  be  a  square.  The 
first  number,  as  you  see,  is  24  ;  this  divided 
by  6  gives  4,  which  is  a  square  (and  its  root  is 
two).  Now  take  the  congruous  number  corre- 
sponding to  24,  which  is  25.  Divide  it  by  4, 
which  is  the  quotient  of  the  first  one,  24, 
divided  by  6,  and  we  have  6,  |,  and  this  is 
the  required  number.  Add  6  to  it  and  you 
have  12,  ^,  which  is  a  square  number,  the  root 
being  3,  1.  If  you  subtract  6  you  have  |  and 
the  root  is  |,  since  a  half  times  a  half  is  i  and 
the  same  number,  6|,  is  a  square  of  which  the 
root  is  two  and  a  half. 
And  so  with  others.* 


25 

24 

100 

96 

169 

120 

225 

216 

289 

240 

400 

384 

625 

336\&6oo 

6ye 

480 

841 

840 

900 

864 

1156 

960 

1225 

1 1 76 

1212 

1080 

1681 

720 

2025 

1944 

2500 

1344  2400 

2602 

2160 

2704 

1920 

2809 

2520 

3025 

2905 

3364 

3360 

3600 

3456 

3221 

1320 

4225 

2026 

*  In  the  list  read  1521  for  1212,2601  for  2602,  3721  for  3221,  2925  for  2905,  and  20 16  for  2026. 

In  the  problem,  since 

^  25  ±  24  =  ,1-2, 

,  25   ,24      ^2 

we  have  -^  ±  —^  =  —  =  y^. 

4        4        4 


43 


€r.@uift(onquinM. 
^Si  quincrcd  ballar  o  tc  fuerc  tJ^inahdado  <iueburquee  tree  nu* 
mcrcd  quadrados  o  mas  i  tcle^  que  /untos  en  vno  ^^gart  Wamcro 
quadnadottoma  el  primer  numcro  quadrado  vr\^He,^y  ^l  qual  ^ 
ta  vno  quedan,8,toma  la  mttad,quadra  ron,i6.t  tflo  te  el  fegundo 
^fuft«f9,T,i6,ro»,25,qulniiPiioquedan,24,totTiald»iltPde8,i2, 
quadra,  (bn,i44>r  ellec©  terccro:  Ti  lo  qtiieres  vet  iun)n,g,r,i  6,  r 
i44,ron,i  69,r^Y5  ^cloe  quak«  es,i3,ccitio  tc  j;5:t  nota  qf 02  tfta 
via  Ic  rodras  jpaser  fninftnltum^ 

IfScftaquilHofT. 
C®  »'S^  ^*^^  '"^  ^^^  ^  ^•^  numcro  quadrado  que  qiiftatidek  oiifii 
(laiidole  fu8  tree  raf  scs^gan  ntlinero  quadradcUcgla  ten  a  n.ie 
tea  aun  numero  congriente  t «««  rt  tiiimero  f  uro  ccnfiriicquadra* 
do  cojrcfpondlentc  t  el  talitumcwr  congtMcme  j^anc  kpoMerifie 
vnidadce  qumds  fpnUe  myS<{Ut  manda  ajiiftar  ©  qmtafT.rctt 
fladuenimicnto  parte  drtimerofiito  cci  gruc  qi46dr$!doco;ict^<ii 

diem  c  y  c\  vltimo  adtf^fftmiieto  quddr*  lo  en  fi  mUm  t  lb  pxmo, 
fcra  elniiniei  0  oemandado  como  terac  pou'fte  tycn.pio. 

^lE^iemplo.  ^. 

«  SToma  ,74  >p:imcrnumcro  congnicntc  parte  le  r6i,?  .^i"/j?„*^ 
rarscoDcniandadao  vienc,8  polios qui^Uo .1^f;"«;^$  ^^1*/^^^^ 
Suo quadrado co^rcfpondicnte  viam^.-i-  ^^"^^'l^'Sr^^^^ 
£Ulcp:odi,tocsIri^Teireea'ern«^^^^^^^^ 

aiuftiidole  0  quirandolcTu8.i,rar$eefera  ^"f/[^f  ,^^.^ 

roqyadradoFfuraT?5ea,4Y-i-1?^l<^^"<^2^"^t^^,^^^^^ 
^»-  que  ^8  mnner 0  quadrido  v  fti  ra r 3  c8.5.o  c|?«C08. 


Square  Numbers 

dL  Fifth  problem 

dLIf  you  wish  to  find  three  or  more  square  numbers  which  added  together  make 
a  square,  take  the  first  odd  square  number  9  ;  subtract  one,  which  gives  8  ;  take 
the  half,  its  square  being  16,  and  this  is  the  second  number.  The  sum  of  9  and 
16  is  25  ;  subtract  one  and  we  have  24;  the  half  of  24  is  12;  the  square  of 
12  is  144,  and  this  is  the  third  number.  If  you  wish  a  proof,  the  sum  of  9  and 
16  and  144  is  169,  of  which  the  root  is  13,  as  you  see.  In  this  way,  you  can  solve 
any  number  of  problems.* 

dL  Sixth  problem 

CLFind  such  a  square  number  that  if  you  subtract  from  it  or  add  to  it  thrice  its 
root,  you  have  a  square  number.  Keep  in  mind  some  congruent  number  and  its 
corresponding  square  congruous  number.  Divide  the  congruent  number  by  the 
number  by  which  you  are  to  multiply  its  roots  when  you  add  or  subtract  them. 
Divide  this  quotient  into  the  corresponding  congruous  square  number  and  square 
the  quotient.  The  result  will  be  the  required  number,  as  you  will  see  by  the 
following  example. 

(^  Example 

<n,Take  24,  the  first  congruent  number.  Divide  it  by  3,  the  number  by  which  you 
are  to  multiply  the  roots,  and  the  quotient  is  8.  Divide  this  8  into  25,  the  corre- 
sponding square  congruous  number,  and  the  quotient  is  3  and  |.  The  square  of 
this  is  9  and  ||,  the  required  number  which,  added  to  or  subtracted  from  thrice 
its  root,  will  be  a  square  as  you  see.  Add  9  and  ||,  which  is  thrice  the  root,  and 
you  have  19  and  ^^,  which  is  a  square  whose  root  is  4  and  |.  If  you  subtract 
thrice  the  root,  you  have  f^,  a  square  number  whose  root  is  5  eighths. t 

dL  Seventh  problem 

d.  Observe  this  :  2  times  2  is  4,  and  3  times  3  is  9  ;  these  numbers  added  together 
make  13.  Now  find  two  other  numbers,  neither  2  nor  3,  which  squared  in  the 
same  way  and  added  together  will  give  the  same  result  1 3. 

*  Take  as  the  first  odd  number  2  «  +  i .   Following  the  directions,  the  first  square  is  4  ^'^  +  4  «  +  i , 
the  second  is  4  «*  +  8  «^  +  4  n'^,  and  the  third  is  the  square  of  2  «*  +  4  ^'^  +  4  «^  +  2  «. 

t  Since  25  —  24  =  1, 

we  have,  multiplying  by  |f,  9II  —  9I  =  f  |,  a  square,  and  similarly  for  25  +  24  =  49. 


45 


^Bufca.  i.numcro0.quadrados"quc  t)agan  mintero  quadrado 
qucteng^raijOircr^ta  lospzimcros  foR.}.K.4,queruafquadradoy 
(on  9.|[«i6i^|unto3d3ciU25.qc8quadnidoirrura?5C8.5*  l^aes 
nora  qucticrtC8.s*immcrQS,qucron.2.|:.5.  1^imcro9.f;.j.i,^, 
qfonlos  p2opucfto8  i^.s.quces  fu  rars  poii  los  conioircjs  figura 
•da  itluegoiimltiplicaciicru5;)i5iendo,3.T»f5ca.j.fon.9.y2*  ve* 
3C8,4.ron.S,  poiiloe  ala  niaiio  ocrccl^a  cl  vno  oebap  x>c\  otro  f  luc 
go  bueluc  a  ocs  ir  pc:  arriba,  2.  vc3cs,  5.  Ton.  6.  po2  aba^o.3*vc3ce. 
.  4#i2.r€ftfi  tl  mcno^  od  maroj.quc  C8»6.t)cj  2.quedan,6.  fo8  c,ua^ 
lee  parte  poz.s  ra^30elo8numero8  p2cpuc(lo8:cladiicn(micnto 
•c5»i -p*  t  cftc  C8  d  vn  numcro  Dcmado  1:  luego  ruina.9.|r.  s,quc  foil 
lo  pjoduto  Ddo8  que  pjlmcro  multipUcalk  fon.i  7*  \os  quales  par 
tepozeLs*^  daducmmicntoes.j-vr^^^-'s  circgundonumcropc 
inadado,l3  pjucua  xicloolcs  q  quadres.  i-fcr?  fiimfmoesa  »^ir^j 
-~^:n  a  mci  ino8  foiui  i  •  ^  que  fumados  juiuos  como  ve  j8  fon  lo5 
mefmoBrij, 


01  t 

>  $ 


f^Cs  vna  quetienedncopcrasconlacquale^puedeperai  tefdc 
vn  tomin  ^fta  quince  pefoa  t  mas.  jE^einando  qut  es  lo  que  pcfa 
cadaviial^nem^noXapnmcra,i,to/]lareguniida  3  toXa*» 
tercera^i,p8,i,toXaquarta,3,p8,3,to.laquiirta.io,jpe.i,tomto 


Questions   Relating  to  Numbers 

dLRule 

Cn^Find  2  numbers  the  sum  of  the  squares  of  which  will  make  a  square  number 
which  has  an  integral  root.  The  first  numbers  are  3  and  4,  for  their  squares  are 
9  and  16,  and  these  added  together  make  25,  the  root  of  which  is  5.  Observe 
that  you  have  5  numbers ;  the  first  are  2  and  3  ;  the  next  are  3  and  4,  the 
proposed  numbers  ;  and  there  is  also  5,  which  is  their  root.  Place  these  num- 
bers as  you  see  in  the  figure  below.  Then  use  cross  multiplication,  saying 
"  3  times  3  is  9,  and  2  times  4  is  8."  Place  these  numbers  at  the  right-hand 
side,  one  under  the  other.  Then  multiply  again  at  the  top,  2  times  3  is  6  ; 
and  underneath,  3  times  4  is  12.  Now  subtract  the  less  from  the  greater,  that 
is,  6  from  12,  and  there  remains  6.  Divide  this  by  5,  the  root  of  the  assumed 
numbers,  and  the  quotient  is  i^,  one  of  the  numbers  required.  Now  add  8  and 
9,  the  products  of  the  first  multiplication,  and  the  sum  is  17.  Divide  this  by  5 
and  the  quotient  is  3|,  and  this  is  the  second  required  number. 

Proof:  The  square  of  i|  is  i  J^ ;  the  square  of  3 1  is  nJi  i  and  these  added 
together,  as  you  see,  make  13.* 


6 

6         3 


17 


12 


02 

1 

I7l3| 

6 

5 

5 

(II,A  man  has  five  weights  with  which  he  can  weigh  from  I  tomin  to  fifteen  or 
more  pesos.  What  is  the  weight  of  each  ?  The  first,  i  tomin ;  the  second, 
3  tomines ;  the  third,  i  peso  i  tomin ;  the  fourth,  3  pesos  3  tomines  ;  the  fifth, 
10  pesos  I  tomin,  t 

*The  equation  «2  ^  ^,2  _  j^  is  indeterminate.  It  is  given  by  Diophantus  (II,  9),  a  late 
Greek  algebraist  of  c.  a.  d.  275.  In  Sir  Thomas  Little  Heath's  edition  of  Diophantus,  second 
edition,  page  145,  Euler's  general  solution  is  given  as  well  as  the  special  solution  leading  to 
Diophantus's  results.  In  the  latter  solution  {x  +  2)^  +  (2  jr  —  3)2  =  13,  whence  x  =  \  and  the  two 
numbers  are  ^-  and  \.   This  solution  is  rather  more  simple  than  the  one  in  the  text. 

t  This  is  the  well-known  Problem  of  the  Weights.  Expressed  in  tomines  the  weights  are 
r,  3,  9,  27,  8i,  a  geometric  progression.  This  solution  requires  that  the  weights  be  placed  on 
either  or  both  pans  of  the  scales.  It  is  evidently  inserted  to  fill  the  page,  having  no  close  con- 
nection with  the  problems  which  immediately  precede  or  follow. 


47 


C@crattaquC(tion» 
ifj.'WSf  d,3,ron  9>M>vc5C8a4/on>i^/urtifl*08fo,i5,qtice©  nil 
iiicro  quadrado  r  fu  rtAX5  C9,5^a  me  otroo,t)08.  numcro8,q  ni  r<a 
3»^^>4>Y  <l»»c  quadriJdo5  c  fi  iticfnioe  r  lo  p2odu3ido  fumado  fca,25 

ftHegla. 
^3Burcfl,2,iiunicro«  que  funto»  lo8  quadrados  en  vno  J^agan  nu# 
mcro  quadrado  que  tenga  rai'5  Dircrcta:toni«,5,t  t>05e  que  (ui  qua 
dradoe  fon,2s,Y,i44_,que  ron,i69,rfl]^3  ©clos  qualcc  ee^i^^pucs 
nora  que  enla  paffada  a  dta  fcmcfante  nmilk,5,fiunierod  f  aqui  tie 
ne8,4,la  caiifa  €8  que^3,^)4,pnnicro8  nunieroe  tiencn  rat5  ^ifcre 
ta  que  co,5,cl  quake  ci  vno  t>elo8.4,i^  firuc  p02el,j,r  cl,4,t)equi 
c  es  ra][5 1 108  otroe  fo  jS.r.ii,  numeroe  |?allado8  opKtfupucflca 
t  fu  ra  v3  que  e8>i3,lo8  quales  pon  enla  inanera  quev  er8  figurado 
t  con  cl,$  ,raf  3,T)e,3,^43p2imeros  numeros  niu].cl,5.T  el.u  ,ntir.e 
roe  pjopueftos  pi3iendo,5,Te3e8,5,fon,^$,t35>v^3^  D05efon,6o 
parte  entramo©  pjodutoe  poi,i3,quc  ee  la  raf  5Delo8  nuuierce  pw 
pucfloG  I  el  aduenlmfento  feran  I08  numeros  que  bufca5:parte,25> 
po^i3>vfencn,i  ifquee8elvnoparte,6o5po2,ij\5ien?«4,  -^ 
que  C8  cl  otro  los  qualee  fx  I08  muUf  plicae  cada  vno  poj  f\  mefmo 
Y  lop2odn5idoruina8  feran  lo8,ij,que  Dcniar.daecomoTCi^e  niul 
enn.i^.e8*},  J'J.multtpUcaenM,  i|-cc.2i/;^/uraalo8K 
fon.25.como  ipci:8  pojia  figura. 


It  ^S 

partidoj,  25    h-ir    ^^\4it 
1}  13 


n   i«i 


Square  Numbers 


dL  Eighth  problem 

(11.3  times  3  is  9,  4  times  4  is  16,  and  the  sum  of  9  and  16  is  25,  which  is  a 
square  number  having  5  for  its  root.  Find  two  other  numbers  the  sum  of  whose 
squares  is  25. 

dLRule 

dLFind  2  numbers  whose  squares  added  together  will  make  a  square  having  an 
integral  root.  Take  5  and  twelve  whose  squares  are  25  and  144  ;  these  added 
together  make  169,  the  root  of  which  is  13.  Observe  that  in  the  previous  problem 
similar  to  this  one  you  had  5  numbers.    Here  you  have  4.    The  reason  is  that 

3  and  4,  the  first  numbers,  have  an  integral  root  which  is  5,  which  is  one  of  the 

4  numbers  here,  and  serves  for  the  3  and  4  of  which  it  is  the  root ;  the  others 
are  5  and  12,  the  assumed  or  presupposed  numbers,  and  their  root  is  13.  Put 
them  down  as  you  see  below ;  then  taking  the  root  derived  from  3  and  4,  the 
first  numbers,  say  "5  times  5  is  25,  and  5  times  12  is  60" ;  divide  each  product 
by  1 3,  which  is  the  root  of  the  sum  of  the  squares  of  the  proposed  numbers,  and 
the  quotients  will  be  the  required  numbers.  Dividing  25  by  13,  the  result  is  iJ|, 
which  is  the  one  part ;  dividing  60  by  1 3,  the  result  is  4,  -^^ ;  square  each  quotient 
and  add  the  results  and  the  sum  will  be  25  as  required.  That  is,  ii|  squared  is 
3-  16  9 '  4'  tV  squared  is  21.  r^y^;  and  their  sum  is  25,  as  you  see  by  the  work.* 


25 
5 

12 
60 


o 

12  28 

13  Divisor         25     1 1||     60 1 4^8^ 

13  13 


*  This  is  simply  a  variant  of  the  seventh  problem.    Euler's  general  solution,  referred  to  on 
page  47,  applies  to  the  equation  x^  +  j^  =/*  +  g^.    If /'=  3  and^  =  4,  as  in  this  case,  the  solution  is 

where  p  and  q  may  have  any  values  whatsoever. 


49 


^auiftioncfloclartcmairoj  tocantcaalalgeb^. 
^jj^zimcra  quilUoii.  ^ 

'fJBamcwiuimeroqua^'ado  qucrcitendo  Ddl,i5^rnr^"^^^^" 
pzopilaraB.  ^^    , 

f;i^(So  que  el  numcro  fca  vm  cofa  ocmediala  cJ  media  cofa  multi 
pltcalacnUl^asc  -Voc3enroa|mlalca,i5*t-T-t>»5fa6, cu2arai?5 
quadi  ada  i  nus  el  medio  dcLi  cofa  ce  rav5  T>el  numcro  oemadado 
)|>)2ucu4  quadra  ra  j;5quadrada  dc.  i6.r  ma5cl  medio  iftla  cofa  q  ce: 
qtro  ^  medio  |pa5e,2o,^-~quc  ca  el  numero  quadrado  oemaudado 
rcaaoel,i5, t  ^qued4n,4,ir  -r  H^c C3  la  ra^S  ^el  p:opnc>. 

ijSegunda  quirt  ion, 
IfiEs  vno  que  fc  ilejta  en  vn  nauio  t  p^egunta  al  maeflrc  que  co  lo 
quct)a>2Daroef[cteclmacftrc  D(3e  que  no  lel^aDelleuar  mas  q 
tlo8  otrod  boluicndocl  paflafero  a  replica r  quanto  feria  el  maellre 
refponde que  t)anoe  fer rantos pefoe que  nuiltiplicados  po: ft  t  a^ 
yuntando  ios  alop^odutpelremanente  rera^i26o«Pcmflndoquan^ 
tooemandaelmaeilrr. 

cr  9\Qo  que  el  dete  fea  vna  cofa  x>c  ^9.  la  mftad  ee  media  cofa  qua 
dralaen(i|?a5e  -~-iD^e3cnroaruntaloa.i26o.l?a5e,i26o,Tr  vnqr 
to  ray 5  oclos  quales  menos  medio  Dela  cofa  ce  el  numero  ocitlada 
do  od  tlcic:reduce,  1 2  60,1  -^  a  quartos  fon  -'^-  la  ray3  es./ 1  ,me 
dios  rclla  cl  medio  oela  Cofa  que  es  medio  quedan>70,mi^dios  que 
ron,;s>7SVtflntocslo  q^cmadaocl  flete:^2ueua  multfplica.^^, 
en  ft  ^)a5e>i2  25,aguutalosco,55,ron,i  260,4  es  numero ^madado 

^iCprccro  quirt  ion. 
^  t3no  vcn<^c  cab?a^no  fe  las  que  fon  mas  oe  que  Ifef  0  vn  mjercf)^ 
te  y  le  picc^unta  quantas  abra  el  vendedo2  refponde  fon  tantas  que 
fi  Lis  muUiplcay 8  en  ft  y  lo  pioduto  quadrtiplars  cl  vltimo  p^odu  ji 
do  rcra.9oooo^^eniandoqudntas  cab2a6  tenia* 


Noteworthy  Problems 

CJ Problems  of  the  Arte  Mayor,  relating  to  algebra 

dL  First  problem 
dLFind  a  square  from  which  if  i5|  is  subtracted  the  result  is  its  own  root. 

CLRule 

(n,Let  the  number  be  cosa  {x).  The  square  of  half  a  cosa  is  equal  to  \  of  a  zenso 
{x^).  Adding  15  and  |  to  \  makes  16,  of  which  the  root  is  4,  and  this  plus  \  is 
the  root  of  the  required  number.* 

Proof:  Square  the  square  root  of  16  plus  half  a  cosa,  which  is  four  and  a  half, 
giving  20  and  \,  which  is  the  square  number  required.  From  20^  subtract  1 5  and  | 
and  you  have  4  and  |,  which  is  the  root  of  the  number  itself. 

(J[  Second  problem 

dl^A  man  takes  passage  in  a  ship  and  asks  the  master  what  he  has  to  pay.  The 
master  says  that  it  will  not  be  any  more  than  for  the  others.  When  the  passenger 
again  asks  how  much  it  will  be,  the  master  replies :  "It  will  be  the  number 
of  pesos  which,  multiplied  by  itself  and  added  to  the  number,  will  give  1260." 
Required  to  know  how  much  the  master  asked.f 

dLRule 

dLLet  the  price  be  a  cosa  of  pesos.  Then  half  of  a  cosa  squared  makes  |  of  a  zenso, 
and  this  added  to  1260  makes  1260  and  a  quarter,  the  root  of  which  less  \  of  a 
cosa  is  the  number  required.  Reduce  1 260  and  \  to  fourths  ;  this  is  equal  to  ^Y"  > 
the  root  of  which  is  71  halves  ;  subtract  from  it  half  of  a  cosa  and  there  remains 
70  halves,  which  is  equal  to  3  5  pesos,  and  this  is  what  was  asked  for  the  passage. 
Proof:  Multiply  35  by  itself  and  you  have  1225  ;  adding  to  it  35,  you  have 
1260,  the  required  number. 

dL Third  problem 

(II.A  man  is  selling  goats.  The  number  is  unknown  except  that  it  is  stated  that 
a  merchant  asked  how  many  there  were  and  the  seller  replied :  "  There  are  so 
many  that,  the  number  being  squared  and  the  product  quadrupled,  the  result  will 
be  90,000."    Required  to  know  how  many  goats  he  had.t 

* x^  —  15I  =  or,  ;r2  —  jr  +  1  =  xd,  x  =  \\^  the  negative  root  being  neglected.  Cosa  (thing) 
was  the  unknown  (x\  and  zenso  (Latin  census)  was  our  x^. 

If  x^  +  X  =  1260,  whence  x  =  ^^.  X  4  x^  —  90,000,  whence  x-  =  1 50. 

51 


^&\^o  que  tcnga  vm  cofa  dc  cab^fls  tmildpKca  en  fl  (Hi3e  w  ten 
fo  tnultiphca  el3enro  po2,4>c(uc  ce  quadruple  \^it  4,5?ftfC8  rgua 
lc8  a.900  oo.cabias  4  c8  numpropanc  numcro  po2  ccnftcladuc 
lumicnto  63,2225 o.rars  t>clo8  qualcs  Ton  las  cab^as  q  tenia*  j^:uc 
ud  toni*i,i  50,  raf  5  oc.22  i$o,niuUipUca  en  fi  (?a5cn,2225o.mHlti< 
plica  loo  po?,4,que  C8  quadruplallo  ron.9oooOk 

^€iuarta  quiflion* 
^.®no  va  po^  pn  camino  p^cguntaa  otro  qucleguas  au{af;>a(fa  v* 
na  cicrta  parte  el  otro  le  rcfpondc  zi  tantas  Icguas  que  fi  lae  inulti 
pllcaf  oenfiylopjoduto  partis  po2,5,claduemimentofcra.8o«t>c*^ 
mandado  que  leguas  sbja  enlo  que  cn3e* 

€|lRegla* 
0[iS>\eo  que  a^^a  \?na  cofa  oe  legua  quadra  la  en  fi  \)fl5c  vn  cefo  par* 
tele  po2»5,  eladuenimicntoes  -Vt)ecenrofGuala.8oJegua8  par 
tcnuineropoKenroqueee.Scpbj  -j-e)  adueniniientocs»4oo.cu 
ta  ra  is  ^on  las  leguas  que  a]?:pues  muiiipHca  ra^3  T)e.4oo4en  fi  q 
es.2o,t  lo  p2odu5ido  parte  po^s^el  aduenimiento  rera.8o.numero 

oemandado. 

C^u^nta  quiflfon. 
CSIno  coinpja  ropa  vtU  tlerra  en  tripla  p2opo:cion  t)e  tal  fuerte  q 
multipUcando  el  triple  poi  el  quarto  t>el fu  triple  que  Ton  las  pie^as 
x>c  ropa  que  conipio  lo pwduto  rcra^48.pero5 Demando  que pieja^ 
t^eropa  comp:o» 

If  J^igo  q«e  conip:o  vna  cofa  i>e  piejfls  T)e  ropa  poj  trcs  cofflc  De 
«8  que  es  en  tripla  mopoKion  dc  ropa  inultiplica  vn  qufli  to  r  c  co- 
fix)cpicvaDeropv!po2,3.corasDe  pefcscs  -^DecenfoFSualcsa 
48.peros  que  es  numero  parte  numero  po2  cefo  que  cs.4b.poJj5^ 
3tdtteniinientocs,64.raHOclosquak8fonlaspicj86Pcropaq 


On  Algebra 


dLRule 

(II,Let  a  cosa  represent  the  number  of  goats.  Squaring  this  we  have  a  zenso ; 
multiplying  the  zenso  by  4,  which  is  a  quadruple,  makes  4  zensos,  which  is  equal 
to  90,000  goats.  Divide  90,000  by  the  number  of  zensos,  and  the  quotient  is 
22,500  [not  22,250  as  given],  the  root  of  which  is  the  number  of  goats  he  had. 
Proof:  Square  150,  the  root  of  22,500,  and  you  have  22,500;  multiply  this 
by  4,  which  is  quadrupling  it,  and  you  have  90,000. 

dL  Fourth  problem 

(II.A  man  traveling  on  a  road  asks  another  how  many  leagues  it  is  to  a  certain 
place.  The  other  replies :  "There  are  so  many  leagues  that,  squaring  the  number 
and  dividing  the  product  by  $,  the  quotient  will  be  80."  Required  to  know  the 
number  of  leagues.* 

(II,Rule 

CLLet  a  cosa  represent  the  number  of  leagues.  This  squared  makes  a  zenso ;  and 
this  divided  by  5  equals  ^  of  a  zenso,  which  is  equal  to  80.  Divide  80  by  ^  and 
the  quotient  is  400,  whose  root  is  the  number  of  leagues  required. 

Proof:  Multiply  the  root  of  400,  which  is  20,  by  itself.    Then  divide  the 
product  by  5  and  the  quotient  is  80,  the  number  required. 

<[L  Fifth  problem 

(II,A  man  buys  a  number  of  pieces  of  clothing  for  three  sums  of  pesos  which  are 
in  triple  proportion,  so  that  multiplying  the  triple  of  the  first  by  ^  of  the  number 
tripled,!  which  is  the  number  of  pieces  of  clothing,  the  product  will  be  48  pesos. 
Required  to  know  how  many  pieces  of  clothing  he  bought.  X 

CILRule 

dl^Let  a  cosa  of  pieces  of  clothing  be  bought  for  three  cosas  of  pesos  in  triple 
proportion  to  the  pieces  of  clothing.  Multiply  a  quarter  of  the  number  of  pieces  of 
clothing  by  3  cosas  of  pesos  and  you  have  |  of  a  zenso,  equal  to  48  pesos.  Divide 
48  by  I  and  the  quotient  is  64,  the  root  of  which  is  the  number  of  pieces  of  clothing 

*  ^x^  =  80,  whence  .r  =  20.  f  That  is,  3  jr  x  ^  ;r.  J  ^  ^*  =  48,  ;jr  =  8. 


S3 


f  OLitp?o  coHaron  Ic  cl  triple  que  to  rar3  tje,676,S^2aeti3  muliipu* 
auu ^ >^ 7^> ^"^ ^^M y^^*^* ^"^ es d  -^ oel fu triple dc rra 
n^^64,cl  Hduemmicnto  c8,483numero  x)cmandada. 

^Sej:ta  quiltioiu 
f;'Slao  ticine  i^eguas  v  vacaa  en  q«(ncuplapx)poK(on  t)C  tdl  fucrtc 
que  fi  inultiplicaa  Usicguas  en  fi  i  las  vacae  en  fv  t  lo  pzoduto  fu-^ 
mad  kmui6g^,x>cmi<io  qntas  fo  tas  teguas^:  quantas  la«  vaca5. 

^Kegla. 
C^^G^  q^e  tenga  t)na  cofa  oc  ireguas  r,5,cora8  T)c  vacas  mulii^ 
plica  ptia  cofacii  li  Ipase  vii  ceiifo  nmL5.cora8  en  ft  |?a3eri;,25,ceros 
aitunta  lo.  ron.26.ccnfo8  Fguales  a,i  664,v'cgua8  ]r  vaca«  nunie^ 
ro^mandado  parte  numcropo2cenroquce8»!664,poi,26,elad'^ 
ucnimicnto  e8,64  ,cu]^a  raf  5  Ton  las  i^eguao  \^  e!  quincuplo  las  va* 
C98  qs  ray3  ;)ea6oo,'jg?iucua  toma  .rars  t)e>64  >^  3^>  qu:ijcupla 
Io8(?a5cii,40,qucesrar5x)e,i6oo7rama€lqih?dradooe,$,  q  fon^ 
las  veguas  con  cl  quadrado  o.e  40,quc  foil  L&  v:iae  t^se^i  664, 
qfacloocmandadc, 

^Sctimaqulilion* 

C^noticnctres  foyaoe  qdrupUpjopoz:ioT>c\)ab2DetaIm.inc 
rj  que  multiplicando  lo  que  vale  la  p^lmera  po?  cl  valo:  ocla  fcgun^ 
dajflopJodusidopojelvalozoeUi  tcrcerael  vltUno  piodutorem^ 
i/aSjOcmando  qucea  el  valojoe  cada  lo^a^ 

€1^egU. 

^»{|o  que  la  pjimcra  valga  vna  cofa  y  U  rcsuda,4,cofiw  f  h  tcr 
cera>i6,quc  conio  vc^s  ellan  en  quadru  pla  p2opo:cfon  mill  vna  co 
fa  poj.4  ^eof^s  cs,4,cenros  multlpUca  lea  poz,i  6  cofas  Dela  terce 
ra  Da5e^,64>^ubo8rgualcsa,i72S,quee8nuniero  partcnunicro 
po2cuboeladuehlmicntoe8,27»cu)?ara?5cubaqueed  ?  eselrai. 
lo2t)dap.imeratUre0undavaleiai3cuba%ea7y.See^^^^ 


Probl 


ems 


that  he  bought.    They  cost  him  three  times  this,  which  is  the  root  of  676  [576]. 

Proof:    Multiply  the  root  of  676  [576],  which  is  24,  by  2,  which  is  \  of  its 

cube,  that  is,  of  8,  the  square  root  of  64.    The  result  is  48,  the  number  required. 

((L  Sixth  problem 

CII.A  man  has  mares  and  cows  in  quintuple  proportion,  in  such  a  way  that  if 
you  square  the  number  of  mares  and  square  the  number  of  cows,  the  products 
added  will  be  1664.    Required  the  number  of  mares  and  the  number  of  cows.* 

CLRule 

dl^Let  there  be  a  cosa  of  mares  and  5  cosas  of  cows.  Squaring  the  first  makes  a 
zenso,  and  5  cosas  squared  makes  25  sensos,  and  the  sum  is  1664  mares  and  cows, 
the  required  number.  Divide  this  number  by  the  number  of  zensos,  that  is,  divide 
1664  by  26,  The  quotient  is  64,  whose  root  is  the  number  of  mares,  and  the 
quintuple,  or  square  root  of  1600,  is  the  number  of  cows. 

Proof:  Take  the  square  root  of  64  ;  it  is  8.  Quintuple  it  and  you  have  40, 
which  is  the  square  root  of  1600.  Add  the  square  of  8,  the  number  of  mares, 
to  the  square  of  40,  the  number  of  cows,  and  you  have  1664,  which  was  required. 

((L  Seventh  problem 

(H^A  man  has  jewels  in  quadruple  proportion  of  value  such  that,  multiplying  the 
value  of  the  first  by  the  value  of  the  second  and  the  product  by  the  value  of  the 
third,  the  last  product  will  be  1728.    Required  the  value  of  each  jewel.t 

C^Rule 

(H,  Let  the  value  of  the  first  be  one  cosa ;  that  of  the  second,  4  cosas ;  and  that 
of  the  third,  16  cosas,  which  you  see  are  in  quadruple  proportion.  Multiply  one  cosa 
by  4  cosas  and  this  is  equal  to  4  zensos.  Multiply  this  by  16  cosas  and  the  result 
is  64  cubes,  and  this  is  equal  to  1728.  Divide  this  number  (that  is,  1728)  by  the 
cube  (that  is,  by  64)  and  the  quotient  is  27,  whose  cube  root  is  3,  the  value  of 
the  first  jewel.    The  second  one  is  worth  the  cube  root  of  1728,  or  12  ;  and  the 

*  .1-2  +  25  j'^  =  1664  (not  1694  as  in  the  original),  whence  jr  =  8. 
f  Take  ;r,  4  x,  xdx.    Then  64  jr^  =  1 728,  whence  jr  =  3. 

55 


tcrccrt  Tale,48,quc  cs  WQ  cuba  rc,23o45|^njc«a  mulcl  taloj  ^ 
la  pjiincra  que  C8,j,po2  cl  ocla  icgunda  que  cs  do3e  t  daduenlmie 
to  po2  la  tcrcera^  que  ce^^SJio  p2ociuto  oclas  muUif^Ucacioneefeni 

>et'auaqu(ftion. 


<|.'!0notienet)i)oett>ffaecnp:opoKfonfisquealtcrat>etaIarteq 
mullos  t>i)06  foi  la»  l^ljas  r  lo  pK^duto  jjoj  la  miwd  tclos  |?i|08  el 
vltimo  p:odu3(do  fera^i  6  2^T)cmando  quantoo  fon  Ico  t^jcd  t  qua-» 
tislaol^ijae. 

IfH^egliu 

ff®ffioquclo8l>(fo0fean  vnaccfa  t  laetil««viiacofat  media 
«ue  e«  en  fiJ  que^Ucra  pK^poklcn  muUwa  cofa  poj  vna  cofa  r  me 
dfa es  vricenfo !!  medio clq««lit»iiU!plica  po2 media ccfaque cs  mi 

t^d  t)elo8  bi)08  t^ses  -^  oeculK)  tguales  a,i62,\?i)08 1^  We  que 
cc  numero  parte  fiumero  pot cubo  que  C8.162.p0j  ^  cUduemml 
cntoe8*2i6.rar5CuteDcfo8qu3le8  rcnIoetljc8t  laatiias  rats 
Cuba  t»e,72^,ieaicua,mulrat3  cuba  t>c,2  i6,que  c5,6,po2  ra^^s  cu 

bat)e-72&4flfc8,9,Ncn54>l5«  <l"«^«  V*^"^^^'"''^'^^^!!* 
qucronlo8jifoabpWutoc6,i62,quec8el  nw^^ 

^T^oueitaquiilioiu 
€®nobat)cba5ert«>0pagamenro8cquar>oplap2opo:d6TJcm 


On  Algebra 


third  is  worth  48,  the  cube  {sic)  root  of  2304. 

Proof:  Multiply  the  value  of  the  first,  which  is  3,  by  that  of  the  second,  which 
is  twelve,  and  this  product  by  the  value  of  the  third,  which  is  48,  and  the  product 
of  the  multiplications  is  1728. 

(H^  Eighth  problem 

dl^A  man  has  a  certain  number  of  sons  and  daughters  in  altera  proportion 
such  that  multiplying  the  number  of  sons  by  the  number  of  daughters  and  the 
product  by  half  of  the  number  of  sons,  the  last  product  will  be  162.  Required 
the  number  of  sons  and  the  number  of  daughters.* 

dLRule 

dLLet  the  number  of  sons  be  a  cosa,  and  the  number  of  daughters  be  a  cosa  and  a 
half,  which  numbers  are  in  altera  proportion.  Multiply  one  cosa  by  a  cosa  and 
a  half  and  the  result  is  a  zenso  and  a  half,  which  multiplied  by  half  a  cosa,  which 
is  half  the  number  of  sons,  makes  |  of  a  cube  which  is  equal  to  162,  the  number 
of  sons  and  daughters.  Divide  this  number  by  the  number  of  cubes,  that  is,  divide 
162  by  |,  and  the  quotient  is  216,  the  cube  root  of  which  is  the  number  of  sons, 
and  the  cube  root  of  729  is  the  number  of  daughters. 

Proof:  Multiply  the  cube  root  of  216,  which  is  6,  by  the  cube  root  of  729, 
which  is  9,  and  the  result  is  54  ;  multiply  this  by  half  of  six,  which  is  the  number 
of  sons,  and  the  result  is  162,  the  number  required. 

(H^ Ninth  problem 

(11, A  man  has  two  payments  to  make,  in  quadruple  proportion  of  months,  so  that 
squaring  the  first,  multiplying  the  product  by  the  quadruple,  and  cubing  this  prod- 
uct, the  result  will  be  32,768.    Required  to  know  how  the  payments  were  made.t 

*  By  the  ancient  Greek  theory  of  proportion  (ratio),  |  x  and  x  are  in  altera  proportion.  The 
word  proportion  was  commonly  used  for  ratio  in  the  sixteenth  century,  and  the  same  is  still  the 
case  outside  the  school.    Take  x  and  |  x.   Then  |  jr^  =  1 62,  whence  x  =  d. 

t  64  jr®  =  32,768,  whence  x  =  2. 


57 


CBt^o  que!o9.2.pasaincnto«  fean  f na  cofa  f  4x0^9  que  e;  cua 
dropla  p^opoKion  quddra  eltb  quadroplo  es  vn  cenfo  mulhplica  le 
po2  el  quadruple  cs*  4*  cuboe  cubicalos  a5e8.64*cubo8t)ecubo8 
rguales  a.  3276S.  quec8numero:  oartenumeropo^  cubosoe 
cuboaqueea.  3276$.  po^64«eladaenimieiitoet.5ii4CU]^a  ra« 
{5  cttbaoe  rar^  cubafcranlod  mefes  ocl  pnmer  pagamento  f  raF5 
cybtca  oe  ra?5  qdrada  oe»262i44.fera  el  fegundo  pagamf  to.^:u 
nia:ra73  cubica  De.5  i2.c8.S*{:  ra^S  cubfca  Dc«S.e9.2.quced  el  pn^ 
incrpU3o:^ra?5  qdrada  t)e.262i44.e8.si2.rraf5cubicape,5i2. 
c«,8.quee8eiregundopU3o:quadracL  2,qc8ru  quadruple  x>c,S. 
€8*4.elqual  muleiplica  po2  d  quadrupb  e8«p.  cubicalo8  el  vltimo 
pN>au3(doe8,3276$.numero  oemandado* 

i5^*e3enaquiflion, 
^tan  J)omb2c  tfenc  ooa  (?ijoJ  en  pjopoKio  fi8  que  qu^a  tl  (^dad 
en  tal  manera  que,mulel  fu  quadruple  oela  M«d  x>ci  inciioi  po?  el 
(Uqufncuplo;>cla  bedao  tJclmatoi  t  lo  que  falicrequadruplando  t 
X)elo  p2odu5ido  facade  fura^  t  cubicado  la  m'itad  cl  vltimo  p^odu 
3(derera.i2$.afto5:ocmandoqHe(?cdadtienecadaviio, 

f:iRegla. 
HiBlgo  que  clmcno2  a^atma  corar)eafio9  f(\  ma)ro2  «faf.t>na 
eofa  Yr^  x>c  cofa  vc  ano8  que  C8  (is  que  qunrra  p20po2ci6,niuLel  fu 
quadrrplooclmeno^poiclfuqulncuplo  t»clma^02quec8  4-poj 
~-lop:odutoe8  rf  ^^<^«^foquadrupla  -~  es  •^t>c  cenfo 
fu  rawedmedia cofa  cubica  fu  mitad  que  ee  vn  qrto t>e cofa  cl  vlti 
mop2Ddu3ldoe$  «-7  0ccuberguala.i2  5,quee5loquefebufcapte 
r*fmcropojcuboquee0.i25,poi  ^^eladucnimienteca.  8000, 
cuf  ^raF5clib(cafonlo8ari08T)el  'meno?^rar 5  quadrada  t)e,625. 
fonlos  oclUiiiyo?.l|^2ucuii  tcma  ra? 3  cuba  t)e«Sooo.bedad  Del  me 
no?e8.2o.mulelfuquadtuploauece»$  poulfuquineuplooe  25. 
raE3  quadrada  De.625.que  e8,5.lo  p2oduto  es^as^quadiuplalos  t)a 


Noteworthy  Problems 


dLRule 
(II,Let  the  2  payments  be  represented  respectively  by  one  cosa  and  4  cosas,  which 
are  in  quadruple  proportion.  Square  the  first,  making  a  zenso ;  multiply  this  by 
the  quadruple,  giving  4  cubes  ;  cube  this,  making  64  cubes  of  cubes  equal  to  32,768. 
Divide  32,768  by  64  and  the  quotient  is  512,  of  which  the  cube  root  of  the  cube 
root  is  the  number  of  months  for  the  first  payment.  The  cube  root  of  the  square 
root  of  262,144  will  be  the  number  of  months  for  the  second  payment. 

Proof:  The  cube  root  of  5 1 2  is  8,  and  the  cube  root  of  8  is  2,  which  is  the 
number  of  months  for  the  first  payment.  The  root  of  262,144  is  512,  and  the 
cube  root  of  512  is  8,  which  is  the  second  number  of  months.  Square  the  2, 
which  is  a  fourth  of  8,  and  this  is  equal  to  4,  which,  multiplied  by  the  quadruple  8, 
is  equal  to  32.    Cube  32,  and  the  result  is  32,768,  the  required  number. 

(J^ Tenth  problem 
(n.A  man  has  two  sons  whose  ages  are  in  such  a  quarta  proportion  that,  multiply- 
ing one  fourth  of  the  age  of  the  younger  by  one  fifth  of  the  age  of  the  elder, 
and  quadrupling  the  result,  and  cubing  half  the  root,  the  final  product  will  be 
125  years.    What  is  the  age  of  each  ?  * 

OLRule 
(n.Let  the  age  of  the  younger  be  a  cosa  of  years  and  that  of  the  elder  be  a  cosa 
and  I  of  a  cosa  of  years,  which  is  a  quarta  proportion.  Multiply  a  fourth  of  the 
younger  by  a  fifth  of  the  elder,  which  is  5^  by  |.  The  product  is  Jg  of  a  zenso. 
Now  quadruple  Jg,  and  this  is  equal  to  \  of  a  zenso,  of  which  the  root  is  half  a 
cosa.  Cube  half  of  this  and  the  product,  ^^  of  a  cube,  is  equal  to  125.  Divide  125 
by  g^j  and  we  have  8000,  whose  cube  root  is  the  number  of  years  in  the  age  of  the 
younger,  and  the  square  root  of  625  is  the  number  of  years  in  the  age  of  the  elder. 
Proof:  The  cube  root  of  8000  is  20,  the  number  of  years  in  the  age  of  the 
younger ;  multiply  a  fourth  of  it,  or  5,  by  a  fifth  of  25,  the  square  root  of  625,  which 
is  5,  and  the  product  is  25  ;  and  this  multiplied  by  4  (that  is,  25  being  quadrupled) 

*  He  means  to  multiply  by  \  instead  of  4,  and  \  instead  of  5.   That  is,  the  age  of  the  younger 

,    .        ,,        5         ^,        X    \     i,x      x^  x^      x^         \x^      X     \    X      X     x^ 

IS  jr;  of  the  elder,  -  x.    Then  -  .  -  .  'i—  =  --•.  ax-—  =  — ;  -y  —  =  -; =  -?  ir-  =  125, 

whencex=2o.         ^  4    S     4        16  16       4      ^4       2     2    2      4     64 


59 


5cndcnto,tomala  mitad^c  furauseeiS.cubicalosclvltimopwdu 
jido  ed.i  25.  nuincro  Dcmandada  lo  qital  nota» 

^Bo  l)cqucndo  fcr  cncfto  maj  largo  Idmopcjeuitarpjolijcidad 
t  lo  otro  po:quc  coino fremp^c  |pc  t>ic^  0  mi  t ntcnto  nunca  fut  otro 
qne  poiicr  Uo  cofaencccffarias  enel  comunt)cfto»rcfgn08:t9fTl 
vcrci^flcoinociiIoDcmasJjcrido  b:cu  c  fupUcoo^/ca  tornado  §  fcr 
uicio  r  rcccbida  la  voluntad  como  9(  quicn  pclTca  feruir. 


On  Algebra 


makes  a  hundred.  Take  half  of  the  root  of  lOO,  which  is  5  ;  cube  this,  and  you 
have  125,  the  number  required,  which  note  carefully. 

dl^It  has  not  been  my  desire  to  extend  this  work,  one  reason  being  that  I  would 
avoid  becoming  tiresome,  and  the  other  reason  being,  as  I  have  always  said,  that  I 
have  wished  merely  to  set  down  the  things  which  are  necessary  and  familiar  in 
this  kingdom.  You  will  therefore  see  that  I  have  always  written  succinctly,  and 
so  I  beg  that  this  book  may  be  judged  and  received  merely  as  the  work  of  one 
who  seeks  to  be  of  service  to  his  fellows. 


61 


finoclaob:a^ 


C  Wyonrcat  glona  t)c  nfo  kfioz  Scfu 

£(^nlto  £  ;)c  la  bzdiu  i  glo^iofa  virgf  fanta  ilfearia  fu  madre 

Ylcfuwa  nra.  ziilq  ft  acaba  el  pfcnte  trawdo  Jetitulado  Su 

mario  cdpcadioTo  oe  cuetae  oe  plata  ^  oeo  nccelTanas  en 

lo8rcr(io3  Dcl]^iru«£lqualfucimp2e(roenla  mu? 

^andctit^i^ne  {:mu|:lcalciudad  oe  J]lbc];ico,tn 

cafa  oc  3}uaii  pablos  15:e(rano :  con  Ucencia  oc( 

inuir  3lluftri(riinoftno2g)on  Xu^sDctac* 

larco,0ifon'C]f  I?  go^cmado^  Delia  IRueua 

€rpana«|£ainnurmocoliceciat>elmu{ 

gUuilrc  i:  reucrcdilTimo.S.Do.frat 

S\\6(oT>c  JOIbotufnT  ar^obifpo  x>c 

inc[:lco:poz  qnto  fue  viilo  t  e):a* 

minado^lt  fe  t;)aUo  fcr  ^uec^o 

roimpntnirTc*  Scabofe  oe 

Im^miniivcpitcrnue 

ueT»U90elme8oe 

i^ifea^o.SInodlna 

cimicto  DC  nfo 

Seftoijefii 

anos 

***** 

*** 

* 


End  of  the  work 
([[To  the  honor  and  glory  of  our  Lord  Jesus  Christ 

and  the  blessed  and  glorious  Virgin  Holy  Mary,  His  mother 
and  our  Lady.    This  is  the  end  of  this  treatise  entitled 
Sumario   copendioso  of  the  computations  of  gold   and 
silver  necessary  in  the  Kingdoms  of  Peru.    This  was 
published   in   the    magnificent,    famous,  and   most 
loyal  City  of  Mexico,  in  the  house  of  Juan  Pablos 
Bressano  ;  with  the  permission  of  the  illustrious 
senor  Don  Luys  de  Velasco,  Viceroy  and 
Governor  of  New  Spain  and,  also,  with 
the  permission  of  the  most  illustrious 
and  reverend  senor,  Brother  Alonso 
de  Montufar,  archbishop  of  Mexico, 
by  whom  this  has  been  seen  and 
examined   and  found  worthy 
to  be  printed.    The  print- 
ing was  finished  on  the 
twenty-ninth    day    of 
the  month  of  May 
in  the  year  of  our 
Lord    Jesus 
Christ 
1556 


63 


INDEX 


PAGES 

Abbreviations  of  monetary  units     .      9-1 1 

Algebra 8,15,51-61 

Alonso  (Alphonso)  de  Montufar      .      3,  63 

Antonio  de  Mendoza 3 

Arte  mayor i5)  51 

Bibliography 5,6 

Books  printed  before  1557     .     .     .     .5,6 

Cardan      . 8 

Census 51 

Common  rules 15 

Congruent  numbers      .     .     .     .  41,  43, 45 
Congruous  numbers     ....  41,43,45 

Congruum 41 

Cortes . 3 

Cosa 51 

Cromberger 3 

Cuento 9 

Fibonacci 41 

Grain 11 

Grano 11 

Juan  Cromberger 3,  5 

Juan  Diez 7 

Juan  Pablos 3)  5>  63 

Juan  de  Zumarraga 3,  5 

Leonardo  Fibonacci 41 

Luys  de  Velasco 3,(>3 

Maravedi 9,11 

Mariani 7 


PAGES 

Mendoza 3 

Mexico  founded 3 

Montufar 3.  63 

Multiplication 29 

Pablos 3)  5)  63 

Paxi 7 

Per  cents 11 

Peso 9 

Printing  in  Mexico 3,  5 

Problem  of  the  weights 47 

Real 9 

Reduction  of  money 23 

Root  for  square  root 37 

Rule  of  Three    ........  39 

Sumario  printed 5,  6 

Tables 9 

Tenochtitlan 3 

Text 13 

Tomin 9,  n 

U  for  1000 II 

University  of  Mexico 3 

Vander  Hoecke 8 

Vara 11 

Velasco 3, 63 

Weights,  problem  of 47 

Zenso 51 

Zumarraga 3,  5 


65 


230  ^Ib'db 


